THE  ELEMENTS 

OF 
ALTERNATING  CURRENTS 


THE   ELEMENTS 


OF 


ALTERNATING   CURRENTS 


BY 


W.  S.  FRANKLIN  AND  R.  B.  WILLIAMSON 

K 


SECOND  EDITION 
REWRITTEN   AND   ENLARGED 


gorfc: 
THE   MACMILLAN    COMPANY 

LONDON  :   MACMILLAN  &  CO.,  LTD. 

IQIO 
All  rights  reserved 


Engineering 
Library 


COPYRIGHT,  1899 
BY  W.  S.  FRANKLIN 

COPYRIGHT,  1901 
BY  W.  S.  FRANKLIN 

Set  up  and  electrotyped  August,  1899.  Reprinted  August,  1900. 
New  Edition,  reset  and  electrotyptd  August,  1901.  Reprinted 
January,  1903 ;  February,  1904 ;  October,  1904 ;  March,  1906 ; 
January,  1907;  March,  1909;  March,  1910. 


PRFRS  OF 

THE  NEW  ERA  PRINTING  COMPJ 
LANCASTER,  PA. 


PREFACE  TO   SECOND   EDITION. 


Two  years'  use  of  Franklin  and  Williamson's  Elements  of  Al- 
ternating Currents,  and  many  valuable  suggestions  from  friends, 
make  it  possible  to  issue  a  greatly  improved  edition,  which  is 
now  offered  to  the  public.  It  is  not  expected  that  this  text  will 
be  found  easy  reading,  but  it  is  certain  that  any  one  of  moderate 
ability  can  gain  by  serious  study  of  this  text  a  clear  understand- 
ing of  the  principles  of  alternating  currents  and  of  the  theory  of 
the  various  types  of  alternating  current  machinery. 

Aside  from  the  minor  changes  which  occur  on  almost  every 
page  of  the  book,  this  new  edition  differs  from  the  old  in  having 
a  very  complete  series  of  practical  problems  with  answers.  The 
alterations  in  the  chapters  on  the  Alternator,  on  the  Transformer, 
on  the  Synchronous  Motor,  on  the  Rotary  Converter  and  on  the 
Induction  Motor,  are  especially  noteworthy. 

The  last  five  chapters  have  been  added  to  give  the  student  a 
general  description  of  some  of  the  ordinary  types  of  alternating 
current  apparatus  and  the  conditions  under  which  it  is  operated. 
These  chapters  are  intended  to  be  read  in  connection  with  the 
corresponding  chapters  that  treat  on  the  theory  of  the  apparatus 
under  consideration. 

The  authors'  thanks  are  due  to  Professor  Morgan  Brooks  for 
many  valuable  suggestions,  to  Messrs.  C.  M.  Crawford  and  H. 
W.  Brown  for  assistance  in  preparing  copy,  in  solving  problems 
and  in  reading  proof,  and  to  the  manufacturing  companies  who 
have  kindly  furnished  a  number  of  the  cuts  used  in  the  chapters 
that  have  been  added.  W.  S.  F. 

R.  B.  W. 

SOUTH  BETHLEHEM, 
July  30,  1901. 

238210 


PREFACE  TO   FIRST   EDITION. 


THIS  book  represents  the  experience  of  seven  years'  teaching 
of  alternating  currents,  and  almost  every  chapter  has  been  sub- 
jected repeatedly  to  the  test  of  class-room  use.  The  authors 
have  endeavored  to  include  in  the  text  only  those  things  which 
contribute  to  the  fundamental  understanding  of  the  subject  and 
those  things  which  are  of  importance  in  the  engineering  practice 
of  to-day. 

It  may  be  taken  for  granted  that  the  authors  are  deeply  in- 
debted to  Mr.  C.  P.  Steinmetz,  whose  papers  are  unique  in  their 
close  touch  with  engineering  realities.  W.  S.  F. 

SOUTH  BETHLEHEM, 
June,  1899. 


vn 


TABLE   OF   CONTENTS. 


CHAPTER    I. 
INDUCTANCE  AND  CAPACITY. 

CHAPTER  11. 

THE  SIMPLE  ALTERNATOR.     ALTERNATING  CURRENT  AND  ALTERNATING  ELEC- 
TROMOTIVE FORCE. 

CHAPTER   III. 

MEASURING  INSTRUMENTS. 

CHAPTER   IV. 
HARMONIC  ELECTROMOTIVE  FORCE  AND  HARMONIC  CURRENT. 

CHAPTER   V. 
PROBLEMS  OF  THE  INDUCTIVE  CIRCUIT  AND  RESONANCE. 

CHAPTER   VI. 
THE  USE  OF  COMPLEX  QUANTITY. 

CHAPTER   VII. 

THE  PROBLEM  OF  COILS  IN  SERIES.     THE  PROBLEM  OF  COILS  IN  PARALLEL. 
THE  TRANSFORMER  WITHOUT  IRON. 

CHAPTER  VIII. 
POLYPHASE  ALTERNATORS.     POLYPHASE  SYSTEMS. 

CHAPTER   IX. 
THE  THEORY  OF  THE  ALTERNATOR. 

CHAPTER   X. 

THE  THEORY  OF  THE  TRANSFORMER. 
ix 


TABLE   OF   CONTENTS. 

CHAPTER  XL 
THE  THEORY  OF  THE  TRANSFORMER. — Continued. 

CHAPTER   XIL 
THE  THEORY  OF  THE  SYNCHRONOUS  MOTOR. 

CHAPTER   XIII. 
THE  THEORY  OF  THE  ROTARY  CONVERTER. 

CHAPTER   XIV. 
THE  THEORY  OF  THE  INDUCTION  MOTOR. 

CHAPTER   XV. 
TRANSMISSION  LINES. 


ALTERNATING   CURRENT    MACHINERY. 

CHAPTER   XVI. 

ALTERNATORS. 

CHAPTER   XVII. 

• 

TRANSFORMERS. 

CHAPTER   XVIII. 
INDUCTION  MOTORS. 

CHAPTER   XIX. 
SYNCHRONOUS  MOTORS. 

CHAPTER   XX. 
ROTARY  CONVERTERS  AND  MOTOR  GENERATORS. 


SYMBOLS. 


i         instantaneous  value  of  current. 
I       maximum  value  of  an  harmonic  alternating  current. 
/        effective  value  of  an  alternating  current. 
e        instantaneous  value  of  electromotive  force. 
E      maximum  value  of  an  harmonic  alternating   electromo- 
tive force. 

E       effective  value  of  an  alternating  electromotive  force. 
r,  R,  resistance  (r  sometimes  used  for  radius). 
L        inductance. 
C       electrostatic  capacity. 
/         time. 
N      turns  of  wire. 

n         speed  in  revolutions  per  second. 
/        frequency  in  cycles  per  second. 
ft>        frequency  in  radians  per  second, 
/i        magnetic  permeability. 
/         length. 
s         sectional  area. 
<l>       magnetic  flux. 
4>t      flux-turns. 
H      magnetic  field  intensity. 
/         square  root  of  minus  one,  \/  —  1. 
x,  X,  reactance. 
s,  Z,  impedance. 
e         base  of  Naperian  logarithms. 


XI 


THE 

ELEMENTS  OF  ALTERNATING  CURRENTS. 


CHAPTER    I, 

INTRODUCTION. 

INDUCTANCE   AND    CAPACITY. 

1.  Magnetic  flux. — Let  s  be  the  area  of  a  surface  at  right 
angles  to  the  velocity  of  a  moving  fluid  and  let  v  be  the  velocity 
of  the  fluid.  Then  sv  is  the  flux  of  fluid  across  the  area  in  units 
volume  per  second.  Similarly,  the  product  of  the  intensity,  H, 
of  a  magnetic  field  into  an  area,  sy  at  right  angles  to  H  is  called 
the  magnetic  flux  across  the  area.  That  is 

&  =  Hs  (i) 

in  which  <I>  is  the  magnetic  flux  across  an  area  s,  which  is  at 
right  angles  to  a  magnetic  field  of  intensity  H. 

The  unit  of  magnetic  flux  is  the  flux  across  one  square  centi- 
meter of  area  at  right  angles  to  a  magnetic  field  of  unit  intensity. 
This  unit  flux  is  called  a  line  of  force  *  or  simply  a  line.  For 
example,  the  intensity  of  the  magnetic  field  in  the  air  gap  between 

*  A  line-of  force  is  a  line  drawn  in  a  magnetic  field  so  as  to  be  in  the  direction  of 
the  field  at  each  point.  The  term  line  of  force  is  used  for  the  unit  flux  for  the  follow- 
ing reason  :  Consider  a  magnetic  field.  Imagine  a  surface  drawn  across  this  field. 
Suppose  this  surface  to  be  divided  \K\.Q  parts  across  each  of  which  there  is  a  unit  flux. 
Imagine  lines  of  force  drawn  in  the  magnetic  field  so  that  one  line  of  force  passes 
through  each  of  the  parts  of  our  surface.  Then  the  magnetic  flux  across  any  area 
anywhere  in  the  field  will  be  equal  to  the  number  of  these  lines  which  cross  the  area. 

I 


.dF'YALTERNATING   CURRENTS. 


the  pole  face  of  a  dynamo  and  the  armature  core  is,  say,  5,000 
units,  and  this  field  is  normal  to  the  pole  face  of  which  the  area 


Fig.  1. 

is  300  square  centimeters,  so  that   1,500,000  lines  of  magnetic 
flux  pass  from  the  pole  face  into  the  armature  core. 

The  trend  of  the  lines  of  force 
near  the  poles  of  a  magnet  is  shown 
in  Fig.  i.  In  Fig.  2  is  shown  the 
trend  of  the  lines  of  force  through 
a  coil  of  wire  in  which  an  electric 
current  is  flowing. 


2.  Induced  electromotive  force. — 

When  a  bundle  of  N  wires  con- 
nected in  series  moves  across  a  mag- 
netic field  so  as  to  cut  the  lines  of  force,  in  each  wire  an  electro- 
motive force  is  induced  which  is  equal  to  the  rate,  d<$>jdt,  at  which 
lines  of  force  are  cut,  and  the  total  electromotive  force  induced  in 
the  bundle  of  wires  is 


Fig.  2. 


N 


dt 


Similarly,  when  the  magnetic  flux  through  a  coil  changes,  an 
electromotive  force  is  induced  in  the  coil,  such  that 


in  which  N  is  the  number  of  turns  of  wire  in  the  coil,  dQjdt  is  the 
rate  of  change  of  flux,  and  e  is  the  induced  electromotive  force. 
The  negative  sign  is  chosen  for  the  reason  that  an  increasing 


INDUCTANCE   AND   CAPACITY.  3 

positive  flux  produces  a  left-handed  electromotive  force  in  the 
coil.* 

Examples.  —  (a)  A  conductor  on  a  dynamo  armature  cuts  the 
1,500,000  lines  of  force  from  one  pole  face  in,  say,  -fa  second, 
that  is,  at  the  rate  of  75,000,000  lines  per  second;  and  this  is 
the  electromotive  force  (in  c.g.s.  units)  induced  in  the  conductor. 

(ft)  A  coil,  having  N  turns  of  wire  surrounds  a  magnet  NS, 
through  which  there  are  <£  lines  of  flux,  as  shown  in  Fig.  I  .  The 
coil  is  quickly  removed  from  the  magnet,  reversed,  and  replaced, 
the  whole  operation  being  accomplished  in  /  seconds.  The  flux 
<f>,  being  reversed  with  respect  to  the  coil,  is  to  be  considered  as 
changing  from  -f  4>  to  —  4>,  the  total  change  being  therefore  2<l>. 
Dividing  this  total  change  of  flux  by  the  time  /  gives  24>//,  which 
is  the  average  value  of  d^ldt  during  the  time  /,  so  that  the  aver- 
age electromotive  force  induced  in  the  coil  is  N-2^jt.  This 
electromotive  force  is  expressed  in  c.g.s.  units,  and  is  to  be 
divided  by  io8  to  reduce  it  to  volts. 

3.  The  magnetic  field  as  a  seat  of  kinetic  energy.  —  The  magnetic 
field  is  a  kind  of  obscure  motion  of  an  all-pervading  medium, 
the  ether  ;  and  this  motion  represents  energy.     The  amount  of 
energy  in  a  given  portion  of  a  magnetic  field  is  proportional  to 
the  square  of  the  intensity  of  the  field.     This  is  analogous  to  the 
fact  that  the  kinetic  energy  of  a  portion  of  a  moving  liquid  is 
proportional  to  the  square  of  the  velocity  of  the  liquid. 

4.  Kinetic  energy  of  the  electric  current  in  a  coil.     Definition 
of  inductance.  —  The  kinetic  energy  of  an  electric  current  is  the 
energy  which  resides  in  the  magnetic  field  produced  by  the  cur- 
rent.    The  kinetic  energy  is,  at  each  point,  proportional  to  the 
square  of  the  field  intensity,  that  is,  to  the  square  of  the  current. 
Therefore  the  total  kinetic  energy  of  the  field  is  proportional  to 
the  square  of  the  current.     That  is 


*  This,  although  an  inadequate  statement,  must  suffice  ;  especially  inasmuch  as  the 
sign  in  equation  (ii)  is  of  no  practical  importance. 


4  ELEMENTS   OF   ALTERNATING   CURRENTS. 

in  which  W  is  the  total  energy  of  a  current  i  in  a  given  coil,  and 
(l/zE)  is  the  proportionality  factor.  The  quantity  L  is  called  the 
inductance  of  the  given  coil. 

Units  of  inductance. — When  in  equation  (i)  J^Fis  expressed  in 
joules  and  i  in  amperes,  then  L  is  expressed  in  terms  of  a  unit 
called  the  henry.  When  W  is  expressed  in  ergs  and  i  in  c.g.s. 
units  of  current,  then  L  is  expressed  in  c.g.s.  units  of  induc- 
tance. The  c.g.s.  unit  of  inductance  is  called  the  centimeter,  for 
the  reason  that  the  square  of  a  current  must  be  multiplied  by  a 
length  to  give  energy  or  work  ;  that  is,  inductance  is  expressed 
as  a  length  and  the  unit  of  inductance  is,  of  course,  the  unit 
of  length.  The  henry  is  equal  to  io9  centimeters  of  inductance. 

Example. — A  given  coil  with  a  current  of  0.8  c.g.s.  unit  pro- 
duces a  magnetic  field  of  which  the  total  energy  is  6,400,000 
ergs,  so  that  the  value  of  L  for  this  coil  is  20,000,000  centime- 
ters. If  the  current  is  expressed  in  amperes  and  energy  in  joules 
then  the  total  energy  corresponding  to  8  amperes  would  be  0.64 
joules  and  the  value  of  L  would  be  0.02  henry. 

Non-inductive  circuits. — A  circuit  of  which  the  inductance  is 
negligibly  small  is  called  a  non-inductive  circuit.  Since  the  in- 
ductance of  a  circuit  depends  upon  the  energy  of  the  magnetic 
field,  a  non-inductive  circuit  is  one  which  produces  only  a  weak 
field,  or  a  field  which  is  confined  to  a  very  small  region.  Thus, 
the  two  wires,  Fig.  3,  constitute  a  non-inductive  circuit,  espe- 


Fig.  3. 

cially  if  they  are  near  together ;  for,  these  two  wires  with  oppo- 
site currents  produce  only  a  very  feeble  magnetic  field  in  the 
surrounding  region.  The  wires  used  in  resistance  boxes  are 
usually  arranged  non-inductively.  This  may  be  done  by  doub- 
ling the  wire  back  on  itself  and  winding  this  double  wire  on  a 
spool.  In  this  case  the  electromotive  force  between  adjacent 
wires  may  be  great  and  they  may  have  considerable  electrostatic 


INDUCTANCE   AND    CAPACITY.  5 

capacity.  In  order  to  make  a  non-inductive  resistance  coil  with- 
out this  defect,  the  wire  may  be  wound,  in  one  layer,  on  a  thin 
paper  cylinder  so  as  to  bring  the  terminals  as  far  apart  as  possible. 
This  cylindrical  coil  is  then  flattened  so  as  to  reduce  the  region 
(inside)  in  which  the  magnetic  field  is  intense.  This  gives  a  non- 
inductive  coil  of  which  the  electrostatic  capacity  is  inconsiderable. 

5.  Moment  of  inertia,  analogue  of  inductance. — The    kinetic 
energy  of  a  rotating  wheel  resides  in  the  various  moving  particles 
of  the  wheel.     The  velocity  (linear)  of  each  particle  of  the  wheel 
is  proportional  to  the  speed  (angular  velocity)  of  the  wheel,  and 
the  energy  of  each  particle  is  proportional  to  the  square  of  its 
velocity,  that  is,  to  the  square  of  the  speed.     Therefore,  the  total 
kinetic  energy  of  the  wheel  is  proportional  to  the  square  of  the 
speed.     That  is, 

W=  y2K«?  (2) 

in  which  IV  is  the  total  energy  of  a  wheel  rotating  at  angular 
velocity  CD  and  (^K)is  the  proportionality  factor.  The  quantity 
K  is  called  the  moment  of  inertia  of  the  wheel. 

6.  Proposition. — The  inductance  of  a  coil  wound  on  a  given  spool 
is  proportional  to  the  square  of  the  number  of  turns  N  of  wire. 
For  example,  a  given  spool  wound  with  No.    16  wire  has   500 
turns  and  an  inductance  of,  say,  0.025   henry;  tne  same  spool 
wound  with  No.   28  wire  would  have  about  ten  times  as  many 
turns  and  its  inductance  would  be  about  100  times  as  great,  or 
2.5  henrys. 

Proof. — To  double  the  number  of  turns  on  a  given  spool  would  everywhere  double 
the  field  intensity  for  the  same  current,  and  therefore  the  energy  of  the  field  would 
everywhere  be  quadrupled  for  a  given  current  so  that  the  inductance  would  be  quad- 
rupled according  to  equation  (i). 

7.  Proposition. — The  inductance  of  a  coil  of  given  shape  is  pro- 
portional to  its  linear  dimensions,  the  number  of  turns  of  wire  be- 
ing unchanged.     For  example,  a  given  coil  has  an  inductance  of 
0.022  henry,  and  a  coil  three  times  as  large  in  length,  diameter, 
etc.,  has  an  inductance  of  0.066  henry. 


6       ELEMENTS  OF  ALTERNATING  CURRENTS. 

8.  Electromotive  force  required  to  make  the  current  in  a  coil 
change.  —  A  current  once  established  in  a  coil  of  zero  resistance 
would  continue  to  flow  without  the  help  of  an  electromotive 
force  to  maintain  it  just  as  a  wheel  when  once  started  continues 
to  turn,  provided  there1  is  no  resistance  to  the  motion  of  the 
wheel.  To  increase  the  speed  of  the  wheel  a  torque  must  act 
upon  it  in  the  direction  of  its  rotation,  and  to  increase  the  cur- 
rent in  the  coil  an  electromotive  force  must  act  on  the  coil  in  the 
direction  of  the  current. 

When  an  electromotive  force  e  (over  and  above  the  electro- 
motive force  required  to  overcome  the  resistance  of  the  coil)  acts 
upon  a  coil  the  current  is  made  to  increase  at  a  definite  rate, 

-.-,  such  that  di 

dt  e  =  L 


Proof  of  equation  (3).  —  Multiplying  both  members  of  this  equation  by  the  current 

i  we  have  ei  =  Li  —  .     Now  ei  is  the  rate,  —  —  ,  at  which  work  is  done  on  the  coil,  in 
at  dt 

addition  to  the  work  used  to  overcome  resistance,  and  this  must  be  equal  to  the  rate 
at  which  the  kinetic  energy  of  the  current  of  the  coil  increases.  Differentiating 

dW  di 

equation  (l)  we  have  —r-  =  Li  —  .     Therefore,  equation  (3)  is  proven. 

Torque  required  to  make  the  speed  of  a  wheel  increase.  —  When 
a  torque  T  (over  and  above  the  torque  required  to  overcome  the 
frictional  resistance)  acts  upon  a  wheel,  then  the  angular  velocity 

flat 
&>  of  the  wheel  is  made  to  increase  at  a  definite  rate,  -~rt  such 

r-ATj  (4) 

Proof  of  equation  (4).  —  Multiplying  both  members  of  this  equation  by  the  angular 

velocity  w  of  the  wheel  we  have  7w  =  Ku  —  .     Now  7w  is  the  rate,  --    at  which 

dt  dt 

work  is  done  on  the  wheel,  and  this  must  be  equal  to  the  rate  at  which  the  kinetic 
energy  of  the  wheel  increases.  Differentiating  equation  (2)  we  have  —--  =J&)  —  • 
Therefore  equation  (4)  is  proven. 

9.  Magnetic  flux  and  flux-turns.  —  In  dealing  with  a  coil  of  wire 


INDUCTANCE   AND   CAPACITY.  7 

it  is  frequently  necessary  to  consider  the  product  of  the  magnetic 
flux  through  the  coil  multiplied  by  the  number  of  tut  ns  of  wire  in 
the  coil.  This  product  is  called  the  flux-turns,  and  it  is  repre- 
sented by  the  single  symbol  4>r  That  is  : 

fc.-A*  (5) 

in  which  4>  is  the  flux  through  the  coil  (strictly  the  flux  through 
a  mean  turn  of  the  coil),  N  is  the  number  of  turns  of  wire  in  the 
coil,  and  <£j  is  the  flux-turns. 

Proposition.  —  The  flux-turns  $>t  through  a  coil  due  to  a  cur- 
rent /  in  the  coil  is 

*,-£.-  (6)* 

in  which  L  is  the  inductance  of  the  coil.  This  proposition  is 
proven  in  the  next  article. 

10.  Self-induced  electromotive  force.  Reaction  of  a  changing 
current.  —  When  one  pushes  on  a  wheel,  causing  its  speed  to  in- 
crease, the  wheel  reacts  and  pushes  back  against  the  hand.  This 

d(d 

reacting  torque  is  equal  and  opposite  to  the  acting  torque,  K-—ry 

[equation  (4)],  which  is  causing  the  increase  of  speed.  Thus, 
when  the  speed  of  the  wheel  is  increasing,  the  reacting  torque  is  in 
a  direction  opposite  to  the  speed,  and,  when  the  speed  is  decreas- 
ing, the  reacting  torque  is  in  the  same  direction  as  the  speed. 

Similarly  when  an  electromotive  force  acts  upon  a  circuit,  f 
causing  the  current  to  increase,  the  increasing  current  reacts. 
The  reacting  electromotive  force  is  equal  and  opposite  to  the  act- 

ing electromotive  force  L  -,-  [equation  (3)]  ,  which  is  causing  the 

current  to  increase.  This  reacting  electromotive  force  is  called  a 
self-induced  electromotive  force.  The  self-induced  electromotive 
force  is  therefore 


*  In  this  equation  L  and  i  must  be  expressed  in  c.g.s.  units  because  the  unit  of  flux 
corresponding  to  the  ampere-henry  is  not  much  used. 

f  Supposed  to  have  zero  resistance  for  the  sake  of  simplicity  of  statement. 


8  ELEMENTS   OF   ALTERNATING   CURRENTS. 

When  a  current  is  increasing  (  -r  positive  j  the  self-induced 
electromotive  force  is  opposed  to  the  current,  and  when  a  cur- 
rent is  decreasing  1  -r  negative  )  the  self-induced  electromotive 

force  is  in  the  direction  of  the  current,  exactly  as  in  the  case  of 
a  rotating  wheel. 

Proof  of  equation  (6).  —  If  the  current  i  in  a  coil  is  changing,  then,  from  equation 
(6)  we  have  d<b^dt—  L  •  di\dt,  and  from  equation  (5)  we  have  d'4>1/flV  =  N-  d$\dt. 
But  —  N'  d$\dt  is  the  electromotive  force,  e,  induced  in  the  coil  by  the  changing  flux, 
or  by  the  changing  current.  Therefore  e  =  —  L-di\dt,  which,  being  identical  to 
equation  (3^),  shows  that  equation  (6)  is  true. 

11,  Calculation  of  inductance  in  terms  of  flux-turns  per  unit 
current.  —  According  to  equation  (6)  the  inductance  of  a  coil  is 
equal  to  the  quotient  3>Ji,  where  4>L  is  the  flux-turns  through 
the  coil,*  due  to  the  current  i  in  the  coil.  There  are  important 
cases  in  which  the  flux  through  a  coil,  due  to  a  given  current, 
may  be  easily  calculated  and,  therefore,  the  inductance  of  such 
a  coil  is  easily  determined. 

Long  solenoid.  —  Consider  a  long  cylindrical  coil  of  wire  of 
radius  r,  of  length  /  and  having  N  turns  of  wire.  The  field  in- 
tensity in  the  coil  is  H  =  qirNijl  and  the  area  of  the  opening  of 
the  coil  is  Trr2,  so  that  the  flux  through  the  opening  is 
(=  <£).  The  coil  has  N  turns,  so  that  <E>X  =  N3>  = 
dividing  this  by  i  we  have,  according  to  equation  (6), 


in  which  L  is  inductance  in  centimeters,  r  is  the  radius  of  the 
coil  in  centimeters  and  /  the  length  of  the  coil  in  centimeters. 
This  equation  is  strictly  true  only  for  very  long  coils  on  which 
the  wire  is  wound  in  a  thin  layer  ;  the  equation  is,  however,  very 
useful  in  enabling  one  to  calculate  easily  the  approximate  induct- 
ance of  even  short  thick  coils. 

*That  is,  the  flux  through  a  mean  turn  multiplied  by  the  number  of  turns  of  wire. 


INDUCTANCE   AND   CAPACITY.  9 

Coil  wound  on  an  iron  core. — A  coil  of  N  turns  of  wire  is  wound 
on  an  iron  ring  /  cm.  in  circumference 
(mean)  and  s  cm.2  in  sectional  area,  as 
shown  in  Fig.  4.  The  coil  produces 
through  the  ring  a  magnetic  flux 
4>  =  m.m f.jm.r.,  where  m.m.f.  (= 
47TiW)  is  the  magnetomotive  force  due 
to  the  coil,  and  m.r.(  —  Ij^is)  is  the 
magnetic  reluctance  of  the  iron  core,  i 
being  the  current  in  the  coil  and  /-t  the 
permeability  of  the  iron.  Therefore,  Fig.  4. 


/ 


Li 


or 


(8) 


Remark. — The  permeability,  /JL,  of  iron  decreases  with  increasing 
magnetizing  force.  Therefore,  the  inductance  of  a  coil  wound  on 
an  iron  core  is  not  a  definite  constant  as  in  case  of  a  coil  without 
an  iron  core. 

12.  Growth  and  decay  of  current  in  an  inductive  circuit. — When 
a  torque  is  applied  to  a  wheel  the  wheel  gains  speed  until  the  whole 
of  the  applied  torque  is  used  to  overcome  the  resistance  of  the 
air,  etc.  While  the  speed  is  increasing  part  of  the  applied  torque 
overcomes  this  resistance  and  the  remainder  causes  the  speed  to 
increase. 

When  an  electromotive  force  is  applied  to  a  circuit  the  current 
in  the  circuit  increases  until  the  whole  of  the  applied  electromo- 
tive force  is  used  to  overcome  the  resistance  of  the  circuit.  While 
the  current  is  growing  part  of  the  applied  electromotive  force 
overcomes  resistance  and  the  remainder  causes  the  current  to  in- 
crease. Therefore, 

di  , 


in  which  E  is  the  applied  electromotive  force,  i  is  the  instantane- 


10  ELEMENTS  'OF   ALTERNATING   CURRENTS. 

ous  value  of  the  growing  current,  R  is  the  resistance  of  the  cir- 
cuit and  L  its  inductance.     Ri  is  the  part  of  E  used  to  overcome 

resistance  and  L-r  is  the  part  of  E  used  to  make  the  current  in- 

crease. 

If  a  circuit  of  inductance,  L,  and  resistance,  R,  with  a  given 
current  is  left  to  itself  without  any  electromotive  force  to  maintain 
the  current,  the  current  dies  away  or  decays,  and  the  electromo- 
tive force  Ri,  which  at  each  instant  overcomes  the  resistance,  is 

the    self-induced    electromotive    force  —  L  -j-  ;    so  that    at  each 

rdi 

instant  Rt  =  —  L  -,-  or 
at 


Examples.  —  An  electromotive  force  of  1  1  o  volts  acts  on  a  coil, 
of  which  the  inductance  is  0.04  henry  and  the  resistance  is  3 
ohms.  At  the  instant  that  the  electromotive  force  begins  to  act, 
the  actual  current  i  in  the  coil  is  zero,  and  the  whole  of  the 
electromotive  force  acts  to  increase  the  current,  so  that  1  10  volts 
=  0.24  henry  x  dijdt  or  dijdt  —  2750  amperes  per  second. 
When  the  growing  current  has  reached  a  value  of  30  amperes, 
Ri  is  equal  to  90  volts,  and  the  remainder  of  the  1  10  volts  acts 
to  cause  the  current  to  increase,  that  is,  20  volts  =0.04  henry  x 
dildt  or  dijdt  =  500  amperes  per  second. 

If  a  current  is  established  in  this  coil  and  the  coil  left  to  itself, 
short  circuited,  without  any  electromotive  force  to  maintain  the 
current  ;  then,  as  the  decaying  current  reaches  a  value  of,  say,  30 
amperes,  the  electromotive  force  Ri  is  90  volts,  and  this  electro  - 

di  di  . 

motive  force  is  equal  to  —  L-,  .  so  that  -j-  is  —  2250  amperes 

per  second. 

13.  Problem  I.  —  An  inductive  circuit  with  a  current  flowing  in  it  is  left  to  itself, 
short  circuited.  At  a  certain  instant,  from  which  time  is  to  be  reckoned  (/=o), 
the  value  of  the  current  is  /.  It  is  required  to  find  an  expression  for  the  decaying 


INDUCTANCE   AND    CAPACITY. 


1 1 


current  at  each  succeeding  instant ;  the  resistance  R  and  the  inductance  L  of  the 
circuit  being  given. 

Let  i  be  the  value  of  the  current  at  the  instant  /.     Then 


Proof. — To  establish  the  truth  of  equation  (ll)  it  is  sufficient  to  show  that  z  — 7 
when  /  =  o,  and  that  equation  ( 10)  is  satisfied.  Substituting  /  =  o  in  equation  (ll) 
we  have  i  =  7.  Differentiating  equation  ( II )  we  have 

di         R--- 


di 


R 


*+*%=* 

which  is  equation  (10). 

The  ordinates  of  the  curve,  Fig.  5,  show  a  decaying  current. 


DECAYING    CURRENT 
=36. Tamp. 


hundredth^  cfagecond 


Fig.  5. 


14.  Problem  II. — A  constant  electromotive  force  E  is  connected  to  a  circuit  of 
resistance  R  and  inductance  Z.  Required  an  expression  for  the  growing  current 
t  seconds  after  the  electromotive  force  is  connected  to  the  circuit. 

The  required  expression  is 

E       E      _R    . 
*  =  -»—-»'*     *    '  <12) 


Proof. — To  establish  the  truth  of  equation  (12)  it  is  sufficient  to  show  that  t'  = 
when  t  =  o  and  that  equation  (9)  is  satisfied. 

The  ordinates  of  the  curve,  Fig.  6,  show  the  values  of  a  growing  current 


12 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


GROWING  CURRENT 
£  =  110  volt? 
H  = 
L  -  0.04- 


lime 


hundredth^  of  a  gecond 


Fig.  6. 

15.  Electric  charge. — The  electric  current  in  a  wire  is  looked 
upon  as  a  transfer  of  electric  charge  along  the  wire.  The  amount 
of  electric  charge  q,  which  in  /  seconds  passes  a  given  point  of 
wire  conveying  a  current  i  is 

g  =  it  (13) 

or  the  rate  dqjdt  at  which  the  charge  passes  a  given  point  on  a 
wire  is 


dq_ 

dt 


04) 


Units  charge. — When  /inequation  (13)  is  expressed  in  am- 
peres and  t  in  seconds,  q  is  expressed  in  terms  of  a  unit  called 
the  coulomb.  That  is,  the  coulomb  is  the  amount  of  electric 
charge  which  passes  in  one  second  along  a  wire  carrying  one 
ampere.  When  i  is  expressed  in  c.g.s.  units  and  /  in  seconds,  q 
is  expressed  terms  of  the  c.g.s.  unit  charge. 

Measurement  of  electric  charge. — An  electric  charge  may  be 
determined  by  measuring  the  current  i  which  it  will  maintain 
during  an  observed  time,  /.  Then  q  may  be  calculated  from  equa- 
tion (13).  The  charge  capacity  of  storage  batteries  is  determined 
in  this  way.  A  very  small  charge  cannot  be  measured  by  meas- 
uring the  current  i  and  the  time  t,  for  such  a  charge  cannot 
maintain  a  steady  measurable  current  for  a  sufficient  time.  A 


INDUCTANCE   AND    CAPACITY.  13 

small  electric  charge  is  measured  by  allowing  it  to  pass  quickly 
through  a  galvanometer  and  observing  the  throw  of  the  needle. 
The  charge  is  sensibly  proportional  to  the  throw.  A  galvanom- 
eter used  in  this  way  is  called  a  ballistic  galvanometer. 

16.  Condensers.  Electrostatic  capacity. — When  the  terminals 
of  a  battery  are  connected  to  two  metal  plates,  as  shown  in  Fig. 
7,  a  momentary  current  flows  as  indi- 
cated by  the  arrows  and  the  electric 
charge  which  passes  along  the  wire 
during  this  momentary  current  is  '  K  • 
stored  upon  the  plates,  for  upon  dis- 
connecting the  battery  and  connecting 
the  plates  with  a  wire  a  momentary 
reversed  current  may  be  observed. 
If  a  ballistic  galvanometer  be  included 

in  the  circuit  the  amount  of  charge  which  passes  into  the  plates  may 
be  measured.  This  amount  of  charge  is  proportional  to  the  elec- 
tromotive force  e  of  the  battery  (other  things  being  equal)  that  is 

q=Ce  (15) 

in  which  q  is  the  electric  charge  which  flows  along  the  wire  into 
the  plates,  e  is  the  electromotive  force  of  the  battery  and  C  the 
proportionality  factor.  Two  plates  arranged  in  this  way  consti- 
tute what  is  .called  a  condenser  and  factor  C  is  called  the  electro- 
static capacity  of  the  condenser.  If,  in  the  equation  (i  5),  q  is  ex- 
pressed in  coulombs  and  e  in  volts,  then  C  is  expressed  in  terms 
of  a  unit  called  a  farad.  That  is,  a  condenser  has  a  capacity  of 
one  farad  when  one  coulomb  of  electric  charge  is  pushed  into  it 
by  a  battery  of  which  the  electromotive  force  is  one  volt.  The 
unit  of  capacity  which  is  commonly  used  to  express  the  capaci- 
ties of  condensers,  electric  cables,  etc.,  is  the  microfarad.  The 
microfarad  is  one  millionth  of  a.  farad.  The  microfarad  is  used 
because  the  farad  is  too  large  a  unit  to  use  conveniently. 

Condensers  to  have  a  larse  canacity  (as  much  as  a  microfarad) 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


are  usually  made  up  of  alternate  sheets  of  tinfoil  and  waxed  papev 
or  mica,  as  indicated  in  Fig.  8.  Alternate  metal  sheets  are  con- 
nected together  as  shown,  thus 
practically  forming  two  plates  of 

large  area. 
=1 

17.  Hydrostatic  analogue  of  the 

condenser. — Consider  a  chamber  with  water-tight  compartments, 
A  and  B,  Fig.  9,  separated  by  an  elastic  diaphragm,  DDy  of  rub- 
ber. If  a  pump  P  is  connected  to  the  com- 
partments as  shown,  a  definite  quantity  of 
water  q  will  be  forced  through  the  pipe,  out 
of  one  compartment  into  the  other,  and  this 
quantity  will  be  proportional  to  the  difference 
of  pressure  e  generated  in  A  and  B  by  the 
pump.  That  is 

q=Ce 

Fig.  9. 

in  which  C  is  the  proportionality  factor.     The 
diaphragm  separating  A  and  B  is  subject  to  mechanical  stress 
very  much  as  the  insulator  or  dielectric  between  the  plates  of  a 
condenser  is  subject  to  electrical  stress. 

18.  Inductivity  of  dielectric. — The  material  between  the  plates 
of  a  condenser  is  called  a  dielectric.  The  capacity  of  a  condenser 
of  given  dimensions  depends  upon  the  material  which  is  used  as 
the  dielectric.  The  quotient,  capacity  of  a  condenser  with  a  given 
dielectric  divided  by  its  capacity  with  air  as  the  dielectric,  is  called 
the  inductivity  of  the  given  dielectric. 

TABLE  OF  INDUCTIVITIES. 
Air  equal  to  unity. 


Glass.  .  .  . 

.    .  3.00  to  10.00 

Mica 

.  .  4.00  to  8.00 

Vulcanite.  . 
Paraffine.  . 
Beeswax.  . 

.    .  2.50 
.    .  1.68  to  2.30 
.    .  1.86 

Shellac    . 
Turpentine  . 
Petroleum   . 

.  .  2.95  to  3.60 
.  .2.15  to  2.43 
.  .  2.04  to  2.42 

INDUCTANCE   AND    CAPACITY. 


The  capacity  of  a  condenser  is  given  by  the  equation  : 


'farads 


ka 
885  x  io-16x  — 


(16) 


!  in  which 

:  turned    in 

body 


in  which  a  is  the  combined  area  in  square  centimeters  of  all  the 
leaves  of  dielectric  between  the  condenser  plates,  x  is  the  thick- 
ness in  centimeters  of  the  dielectric  leaves,  and  k  is  the  inductiv- 
ity  of  the  dielectric  used. 

19.  Mechanical  and  electrical  analogies. — The  analogy  between  moment  of 
inertia  and  inductance  as  pointed  out  in  the  discussion  of  inductance  is  but  a  small 
part  of  an  extended  analogy  between  pure  mechanics  and  electricity.  This  extended 
analogy  is  here  briefly  outlined. 

9  =  »  (3) 

in  which  q  is  the  electric 
charge  which  in  /  seconds 
flows  through  a  circuit  car- 
rying a  current  /. 

W=Eq  (6) 

in  which  W  is  the  work 
done  by  an  electromotive 
force  E  in  pushing  a  charge 
q  through  a  circuit. 

P=Ei  (9) 
in  which  P  is  the  power 
developed  by  an  electro- 
motive force  E  in  pushing 
a  current  i  through  a  circuit. 

in  which  W  is  the  kinetic 
energy  of  a  coil  of  induc- 
tance L  carrying  a  current  i. 

,di 


*==*  (I) 

in  which  x  is  the  distance 
traveled  in  t  seconds  by  a 
bod^  moving  at  velocity  v. 

W=Fx  (4) 
in  which  W  is  the  work 
done  by  a  force  F  in  pull- 
ing a  body  through  the  dis- 
tance x. 

P=Fv  (7) 
in  which  P  is  the  power 
developed  by  a  force  F  act- 
ing upon  a  body  moving  at 
velocity  v. 

W=y2mv*  (10) 
in  which  W  is  the  kinetic 
energy  of  a  mass  m  moving 
at  velocity  v. 

F=m^       (13) 


*  =  6*  (2) 

*   is  the   angle 
/  seconds  by  a 
turning    at    angular 
velocity  w. 

W=T$  (5) 
in  which  W  is  the  work 
done  by  a  torque  T  in  turn- 
ing a  body  through  the 
angle  0. 

P=  Tu  (8) 
in  which  P  is  the  power 
developed  by  a  torque  T 
acting  on  a  body  turning  at 
angular  velocity  «. 

W=#X&  (II) 
in  which  W  is  the  kinetic 
energy  of  a  wheel  of  mo- 
ment of  inertia  K  turning 
at  angular  velocity  u. 


in  wnicn  /•  is  the  force  re- 
quired to  cause  the  velocity 
of  a  body  of  mass  m  to  in- 

dv 

crease  at  the  rate  — 
dt 


(16) 


in  which  T  is  the  torque 
required  to  cause  the  angu- 
lar velocity  of  a  wheel  of 
moment  of  inertia  K  to 

du 

increase  at  the  rate  — 
at 


in  which  E  is  the  electro- 
motive force  required  to 
cause  a  current  in  a  coil  of 
inductance  L  to  increase  at 

di 

the  rate  - 
dt 

q=CE          (18) 


i6 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


— F 


VISE 


BODY 


Fig.  a. 

A  body  of  mass  m  is  sup- 
ported by  a  flat  spring  6", 
clamped  in  a  vise  as  shown 
in  Fig.  a.  A  force  F  push- 
ing sidewise  on  m  moves  it 
a  distance  jr,  which  is  pro- 
portional to  F,  according 
to  equation  (16).  When 
started  the  body  m  will  con- 
tinue to  vibrate  back  and 
forth  and  the  period  r  of  its 
vibrations  is  determined  by 
equation  (19). 


Fig.  b. 

A  body  of  moment  of  in- 
ertia J£  is  hung  by  a  wire 
as  shown  in  Fig.  b.  A 
torque  T acting  on  the  body 
will  turn  the  body  and  twist 
the  wire  through  an  angle 
0,  which  is  proportional  to 
T,  according  to  equation 
(17).  When  started,  the 
body  will  vibrate  about  the 
wire  as  an  axis  and  the 
period  T  of  its  vibrations 
is  determined  by  equation 

(20). 


Fig.  c. 

A  condenser  C  is  con- 
nected to  the  terminals  of 
a  coil  of  inductance  L  as 
shown  in  Fig.  c.  An  elec- 
tromotive force  E  acting 
anywhere  in  the  circuit 
pushes  into  the  condenser 
a  charge  q,  which  is  pro- 
portional to  Ey  according 
to  equation  (18).  When 
started  the  electric  charge 
will  surge  back  and  forth 
through  the  coil,  constitut- 
ing what  is  called  an  oscil- 
latory current  and  the  period 
of  one  oscillation  is  deter- 
mined by  equation  (21). 


PROBLEMS. 

1.  The  intensity  of  the  magnetic  field  in  the  air  gap  between 
the  pole  face  and  the  armature  core  of  a  dynamo  is  5,000  c.g.s. 
units  and  the  pole  face  is  10  cm.  x  20  cm.      Required  the  mag- 
netic flux  from  pole  face  to  armature  core.     Ans.  1,000,000  lines. 

2.  A  coil  of  an  alternator  armature  has  20  turns  of  wire  and 
engages  the  whole  1,500,000  lines  which  flow  from  a  pole  of  the 
field  magnet.     In  ^^  second  this  coil  moves  from  a  north  pole 
to  an  adjacent  south  pole  of  the  field  magnet,  when  the  flux  is 


INDUCTANCE   AND    CAPACITY.  I/ 

reversed.      Calculate  the  average  electromotive  force  in  the  coil 
during  this  interval.     Ans.  1 50  volts. 

3.  The  core  of  an  induction  coil  when  magnetized  has  a  flux 
of  1 20,000  lines  passing   through  it.     When   the  primary  circuit 
is  broken  the  flux  through  the  core  drops  to  15,000  lines  in  -g-J-g- 
second.     What  is  the  average  value  during  this  interval  of  the 
electromotive  force  which  is  induced  in  the  secondary  coil  which 
has  150,000  tufns  of  wire?     Ans.  78,750  volts. 

4.  Find  the  approximate  inductance,  in  henrys,  of  a  cylindrical 
coil  25  cm.  long  and  5  cm.  mean  diameter,  wound  with  one  layer 
of  wire  containing  150  turns.     Ans.  0.00022  henry. 

5.  Calculate  the  kinetic  energy  of  a  current  of  20  amperes  in 
the  above  coil.     Ans.  0.044  joule. 

6.  The  above  coil  is  connected  to  I  ro-volt  mains.     Find  the 
rate,  in  amperes  per  second,  at  which  the  current  begins  to  in- 
crease in  the  coil.     Ans.  500,000  amperes  per  second. 

7.  Calculate  the  rate  at  which  the  current  (problem  6)  is  in- 
creasing when  it  has  reached  the  value  of  10  amperes,  the  resist- 
ance of  the  coil  being  2.5   ohm.     Ans.  386,262.6  amperes  per 
second. 

8.  A  coil  of  which  the  resistance  is  2.5  ohms  and  the  induct- 
ance 0.04  henry  has  a  current  started  in  it.     The  coil  is  then 
short  circuited  and  the  current  left  to  die  away.      Calculate  the 
rate,  in  amperes  per  second,  at  which  the  current  is  decreasing  as 
it  passes  the  value  of  10  amperes.     Ans.  —625    amperes  per 
second. 

9.  A  coil  of  wire  has  an  inductance  of  0.035  henry.      Calculate 
the  magnetic  flux-turns  through  the  coil  due  to  a  current  of  5 
c.g.s.  units  in  the  coil.     The  coil  has  1,500  turns  of  wire  ;  calcu- 
late the  number  of  lines  of  flux  through  a  mean  turn.     Ans. 
175,000,000  line-turns,  116,666  lines. 

10.  A  condenser  has  a  capacity  of  1.2  microfarads.      Calculate 
the  charge  which  is  pushed  into  this  condenser  by  an  electro- 
motive force  of  1,000  volts,  and  calculate  the  time  during  which 


1 8  ELEMENTS   OF   ALTERNATING   CURRENTS. 

this   charge   would   maintain   a   current  of  one   ampere.     Ans. 
0.0012  coulomb,  0.0012  second. 

11.  A  condenser  is  built  up  of  leaves  of  mica,  each  o.  I   milli- 
meter thick,  between  sheets  of  tinfoil  7  x  10  centimeters.      Find 
the  number  of  mica  leaves  required  to  give  a  capacity  of  one 
microfarad.  .  (The  inductivity  of  mica  may  be  taken  as  6.)     Ans. 
269  leaves.. 

12.  A  condenser,  consisting  of  two  sheets  of  tinfoil  30  x  30 
centimeters  pasted  on  the  two  sides  of  a  pane  of  glass   2  milli- 
meters thick  (inductivity  of  glass  equals  6),  is  discharged  through 
a  coil   consisting  of  250  turns  of  wire  wound  on  glass   tube   3 
centimeters  in  diameter  and  30  centimeters  long.     Find  the  ap. 
proximate  periodic  time  of  the  electrical  oscillations.     (See  Arti. 
cle  19.)     Ans.  0.00000417  second. 

13.  *  The  field  coils  of  a  shunt  dynamo  have  a  resistance  of 
100  ohms  and  an  inductance  of  20  henrys.     An  electromotive 
force  of  500  volts  is  applied.      Calculate  the  time  required  for  the 
current  to  reach  a  value  of  4  amperes.     Ans.  0.322  second. 

14.  A   telegraph   line  has  3.6  henrys  inductance  and   2,500 
ohms  resistance.      Calculate  the  time  required  for  the  current  to 
reach  %  of  its  full  value,  E/R,  after  the  circuit  is  closed.     Ans. 
0.00157  second. 

*  This  problem  applies  to  the  case  in  which  the  field  magnet  is  laminated.  When 
the  field  magnet  is  made  of  solid  iron  the  eddy  currents  produced  in  it  during  mag- 
netization permit  the  current  in  the  field  coils  to  grow  very  quickly  to  nearly  its  full 
value  ;  while,  at  the  same  time,  the  magnetization  of  the  core  grows  more  slowly  than 
when  the  iron  is  laminated. 


CHAPTER    II. 
THE   SIMPLE   ALTERNATOR. 

20.  The  alternator.  Definition  of  alternating  electromotive  force 
and  of  alternating  current. — The  alternator  is  an  arrangement  by 
means  of  which  mechanical  energy  or  work  is  used  to  cause  the 
magnetic  flux  from  a  magnet  to  pass  through  the  opening  of  a 
coil  of  wire  first  in  one  direction  and  then  in  the  other  direction. 
This  varying  magnetic  flux  induces  an  electromotive  force  in  the 
coil  first  in  one  direction  and  then  in  the  other  direction.  This 
electromotive  force,  called  an  alternating  electromotive  force y  pro- 
duces an  alternating  current  in  the  coil  and  in  the  circuit  which  is 
connected  to  the  terminals  of  the  coil. 

Examples. — In  the  common  type  of  alternator  the  above-men- 
tioned magnet  and  coil  move  relatively  to  each  other.  Fig.  10 
shows  the  essential  features  of  such  an  alternator.  The  poles  of 
a  multipolar  magnet,  called  the  field  'magnet^  project  radially 
inwards  toward  the  passing  teeth  of  a  rotating  mass  of  laminated 
iron  A,  and  upon  these  teeth  are  wound  the  coils  of  wire  in  which 
the  alternating  electromotive  force  is  induced.  This  rotating 
mass  of  iron  with  its  windings  of  wire  is  called  the  armature. 
On  the  armature  shaft,  at  the  one  end  of  the  armature,  are 
mounted  two  insulated  metal  rings  called  collecting  rings.  These 
metal  rings  are  connected  to  the  ends  of  the  armature  wire,  and 
metal  brushes  rub  on  these  rings,  thus  keeping  the  ends  of  the 
armature  winding  in  continuous  contact  with  the  terminals  of  the 
external  circuit  to  which  the  alternator  supplies  alternating  cur- 
rent. The  electromotive  forces  induced  in  adjacent  armature 
coils  are  in  opposite  directions  at  each  instant  and  the  coils  are 
so  connected  together  that  these  electromotive  forces  do  not  op- 

19 


20 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


pose  each  other.  This  is  done  by  reversing  the  connections  of 
every  alternate  coil,  as  indicated  by  the  dotted  lines  connecting 
the  coils  in  Fig.  10.  The  electromagnetic  action  of  this  type,  of 
alternator  depends  only  upon  the  relative  motion  of  field  magnet 


Fig.  10. 


and  armature,  and  large  machines  are  often  built  with  station- 
ary armature  and  rotating  field  magnet. 

In  another  type  of  alternator,  called  the  inductor  alternator,  the 
magnet  and  the  wire  in  which  the  alternating  electromotive  force 
is  induced  are  both  stationary,  and  a  moving  mass  of  laminated 
iron  causes  the  magnetic  flux  from  the  magnet  to  pass  through 
the  stationary  coils  in  the  desired  manner.  Fig.  1 1  shows  the 
essential  features  of  the  inductor  alternator.  NSNS,  etc.,  are  the 
field  magnet  poles.  The  armature  wire  is  wound  on  the  inter- 
mediate projections  AAA,  and  the  inductors  ///  are  supported 
by  a  spider  keyed  to  a  rotating  shaft. 


THE   SIMPLE   ALTERNATOR. 


21 


The  exciter. — The  field  magnet  of  an  alternator  is  usually  an 
electromagnet  which  is  excited  by  a  continuous  electric  current 
supplied  by  an  indepen- 
dent generator,  generally 
by  an  auxiliary  continuous 
current  dynamo  called  the 
exciter.  The  exciting  cur- 
rent flows  through  coils  of 
wire  wound  on  the  project- 
ing poles  NSNS  in  Figs. 
10  and  1 1. 

Armature  cores  and  ar- 
mature windings.  —  The 
type  of  armature  core 
shown  in  Fig.  10  is  called 
the  toothed  armature  core, 
and  the  winding  is  said  to 

be  concentrated,  that  is,  the  armature  conductors  are  grouped  in  a 
few  heavy  bunches.  Armature  cores  are  also  made  with  many 
small  slots,  in  which  the  armature  conductors  are  grouped  in  small 
bunches.  This  type  of  core  is  called  a  multi-slotted  core,  and  the 

winding  is  said  to  be  distributed. 
The  various  types  of  armature  wind- 
ings are  described  in  Chapter  IX. 

In  some  of  the  earlier  types  of 
alternators  the  armature  core  con- 
sisted of  a  smooth,  cylindrical  mass 
of  laminated  iron,  upon  the  face  of 
which  the  conductors  were  arranged 
in  bands  side  by  side,  one  layer  or 
more  in  depth.  This  type  of 
armature  is  called  the  smooth  core 

Fig.  12. 

armature. 

The  hydraulic  analogue  of  the  alternator. — Consider  a  valveless 
pump  P,  Fig.  1 2,  of  which  the  piston  is  pushed  rapidly  back  and 


22  ELEMENTS   OF   ALTERNATING   CURRENTS. 

forth.  This  to-and-fro  motion  of  the  piston  produces  an  alter- 
nating hydrostatic  pressure-difference  between  the  outlet  and  in- 
let of  the  pump,  and  causes  the  water  to  surge  back  and  forth 
through  the  circuit  of  pipe. 

21.  Advantages  and  disadvantages  of  alternating  currents. — 
The  electric  transmission  of  a  given  amount  of  power  may  be 
accomplished  by  a  large  current  at  low  electromotive  force,  or 
by  a  small  current  at  high  electromotive  force.  In  the  first  case 
very  large  and  expensive  transmission  wires  must  be  used,  or  the 
loss  of  power  in  the  transmission  line  will  be  excessive.  In  the 
second  case  comparatively  small  and  inexpensive  transmission 
wires  may  be  used.  Thus  it  is  a  practical  necessity  to  employ 
high  electromotive  forces  in  long-distance  transmission  of  power. 

High  electromotive  forces  are  dangerous  under  the  conditions 
which  ordinarily  obtain  among  users  of  electric  light  and  power, 
and  many  types  of  apparatus,  such  as  incandescent  lamps,  oper- 
ate satisfactorily  only  with  medium  or  low  electromotive  forces. 
Therefore,  means  must  be  provided  at  a  receiving  station  for 
transforming  the  power  which  is  delivered,  from  high  electromo- 
tive force  and  small  current,  to  low  electromotive  force  and  large 
current  if  long-distance  transmission  is  to  be  successful.  This  is 
called  step-down  transformation.  The  advantage  of  alternating 
current  over  direct  current  lies  almost  wholly  in  the  cheapness 
of  construction,  the  cheapness  of  operation  and  the  high  effi- 
ciency of  the  alternating  current  as  compared  with  the  direct- 
current  apparatus  that  is  required  for  transformation. 

In  step-down  transformation  of  direct  current  a  motor  takes  a 
small  current  from  the  high  electromotive  force  transmission 
mains,  and  drives  a  dynamo  which  delivers  large  current  to  ser- 
vice mains  at  low  electromotive  force.  This  apparatus,  or  its 
equivalent,  the  dynamotor,  is  expensive  to  construct,  it  requires 
attention  in  operation,  and  its  efficiency  is  never,  perhaps,  above 
90  per  cent. 

The  step-down  transformation  of  alternating  currents  is  ac- 
complished by  means  of  the  alternating  current  transformer, 


THE   SIMPLE   ALTERNATOR.  23 

which  is  described  in  Chapter  X.  The  alternating  current  trans- 
former is  very  much  cheaper  than  a  dynamo  and  motor  of  the 
same  output,  it  requires  no  attention  in  operation,  and  its  effi- 
ciency under  full  load  is  usually  greater  than  97  per  cent. 

Alternating  current  has  some  minor  advantages  over  direct 
current,  on  account  of  the  fact  that  alternating  current  machines 
are  frequently  simpler  in  construction  than  direct  current  ma- 
chines. In  particular,  the  commutator  is  not  an  essential  part 
of  an  alternating  current  machine.  In  the  case  of  the  inductor 
alternator  and  the  induction  motor,  the  rotating  part  may  not 
have  any  sliding  electrical  contacts  whatever. 

The  simple  alternating  current  is  not  suited  to  motor  service, 
for  the  alternating  current  motor  does  not  start  satisfactorily. 
For  uninterrupted  service  the  synchronous  motor  is  frequently 
used,  the  starting  being  effected  by  an  auxiliary  engine  or  other 
independent  mover.  The  synchronous  motor  is  described  in 
Chapter  XII. 

The  synchronous  motor  is  not  satisfactory  when  frequent  start- 
ing is  necessary.  For  such  service  the  induction  motor  is  used. 
The  induction  motor  is  described  in  Chapter  XIV.  The  induc- 
tion motor,  to  start  satisfactorily,  must  be  supplied  with  two  or 
more  distinct  alternating  currents,  transmitted  to  the  motor  over 
separate  lines.  This  is  called  the  polyphase  system  of  transmis- 
sion. It  is  described  in  Chapter  VIII. 

For  some  purposes,  especially  for  the  electrolytic  processes, 
which  are  used  on  a  large  scale  in  electro-chemical  works,  direct 
current  only  can  be  used.  When  power,  transmitted  by  alter- 
nating current,  is  to  be  delivered  in  the  form  of  direct  current, 
the  conversion  is  effected  by  means  of  the  rotary  converter,  which 
is  described  in  Chapter  XIII. 

The  rectifier  is  sometimes  used  for  converting  alternating  cur- 
rent into  direct  current.  The  rectifier  is  described  in  Article  28. 
The  rectifier  is  used  in  American  practice  only  in  connection  with 
the  compounding  of  the  field  of  the  alternator  as  described  ia 
Article  94. 


24  ELEMENTS    OF   ALTERNATING   CURRENTS. 

22.  Characteristic  features  of  alternating-current  problems. — 

Alternating-current  problems  differ  from  direct-current  prob- 
lems chiefly  for  two  reasons,  as  follows  :  (a)  The  rapid  changing 
of  the  alternating  current  produces  an  electromotive  force  reac- 
tion in  any  circuit  which  has  inductance,  and  a  portion  of  the 
electromotive  force  which  acts  upon  the  circuit  is  necessarily  used 
in  overcoming  this  reaction.  (<£)  The  rapid  changing  of  an  alter- 
nating electromotive  force  causes  the  various  parts  of  a  circuit  to 
become  alternately  charged  and  discharged  with  electricity,  and 
a  portion  of  the  alternating  current  which  is  delivered  to  a  cir- 
cuit does  not  flow  through  the  entire  circuit,  but  charges  and  dis- 
charges the  various  parts  of  the  circuit.  In  short,  alternating- 
current  phenomena  differ  from  direct-current  phenomena  because 
of  the  effects  of  inductance  and  capacity. 

In  many  of  the  practical  problems  in  alternating  currents  the 
capacity  is  concentrated  at  one  part  of  the  circuit  in  the  form  of 
a  condenser.  These  problems,  which  are  comparatively  simple, 
are  treated  in  Chapters  V.,  VI.  and  VII.  In  the  most  general 
case  the  capacity  is  distributed  throughout  the  circuit,  or  in  other 
words  all  portions  of  the  circuit  are  charged  and  discharged  per- 
ceptibly as  the  alternating  electromotive  force  which  acts  upon 
the  circuit  pulsates.  The  phenomena  exhibited  by  long  trans- 
mission lines  depend  very  materially  upon  the  effects  of  distrib- 
uted capacity.  These  phenomena  are  discussed  in  Chapter  XV. 
A  clear  idea  of  the  effects  of  distributed  capacity  may  be  obtained 
by  considering  Fig.  1 2.  Imagine  the  circuit  of  pipe  in  this  figure 
to  be  made  of  a  distensible  rubber  tube.  Then  at  a  given  instant 
the  flow  of  water  through  the  tube  at  one  point  will  not  be  the 
same  as  the  flow  through  the  tube  at  another  point ;  and  the 
difference  of  flow  at  the  two  points  will  be  accommodated  by  ex- 
pansion and  contraction  of  the  intervening  portion  of  the  tube. 

23.  Speed  and  frequency. — The  electromotive  force  of  an  alter- 
nator passes  through  a  set  of  positive  values,  while  a  given  coil 
of  the  armature  is  passing  from  a  south  to  a  north  pole  of  the 


THE   SIMPLE   ALTERNATOR.  25 

field  magnet,  and  through  a  similar  set  of  negative  values,  while 
the  coil  is  passing  from  a  north  pole  to  a  south  pole,  or  vice 
versa.  The  complete  set  of  values,  including  positive  and  nega- 
tive values,  through  which  an  alternating  electromotive  force  (or 
alternating  current)  repeatedly  passes,  is  called  a  cycle.  The 
number  of  cycles  per  second  is  called  the  frequency,  f. 

Let/  be  the  number  of  pairs  of  field  magnet  poles,  n  the  revo- 
lutions per  second  of  the  armature,  and  /  the  frequency  of  the 
electromotive  force  of  the  alternator.  Then 

f  =  pn  (17) 

This  is  evident  when  we  consider  that  the  electromotive  force 
passes  through  a  complete  cycle  of  values  while  an  armature 
tooth  is  passing  from  a  north  pole  to  the  next  north  pole,  so 
there  are  /  cycles  of  electromotive  force  for  each  revolution  of 
the  armature. 

24.  Electromotive  force  and  current  curves. — The  successive 
instantaneous  values  of  the  electromotive  force  of  an  alternator 
may  be  represented  by  the  ordinates  of  points  on  a  curve,  the 
abscissas  representing  time  elapsed  from  some  chosen  epoch ;  the 
resulting  curve  is  called  the  electromotive  force  curve  of  the  alter- 
nator. In  a  similar  manner  the  successive  instantaneous  values 
of  an  alternating  current  may  be  represented  by  ordinates  and  the 
elapsed  times  by  abscissas  giving  a  current  curve.  These  curves 
are  determined  with  the  help  of  the  contact-maker  as  explained 
in  Article  30. 

Examples. — The  full-line  curve,  Fig.  13,  represents  the  electro- 
motive force  of  an  alternator  with  a  distributed  armature  wind- 
ing ;  and  the  dotted  curve  represents  the  current  which  this  elec- 
tromotive force  produces  in  a  non-inductive  circuit.  This  current 
is  at  each  instant  equal  to  the  electromotive  force  divided  by  the 
resistance  of  the  circuit,  so  that  the  current  is  a  maximum  when 
the  electromotive  force  is  a  maximum.  The  current  is  then  said 
to  be  in  phase  with  the  electromotive  force,  as  is  explained  in 
Chapter  IV. 


26 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


Fig.  13. 


The  full-line  curve,  Fig.  14,  represents  the  electromotive  force 
of  an  alternator  with  a  distributed  armature  winding,  and  the 


Fig.  14. 


dotted  curve  represents  the  current  which  this  electromotive 
force  produces  in  an  inductive  circuit.  In  this  case  part  of  the 
electromotive  force  is,  at  each  instant,  used  to  cause  the  current 


7 


Fig.  15. 


THE   SIMPLE   ALTERNATOR. 


to  increase  or  decrease.     The  part  so  used  is  L  -^  according  to 

equation  (3),  and  the  remainder,  equal  to  Ri,  is  used  to  overcome 
the  resistance  of  the  circuit.  When  the  current  is  zero  then  all 
the  electromotive  force  is  used  to  cause  the  current  to  change, 

since  Ri  is  zero.     When  -y-  is  zero,  the  current  is  at  its  maximum 

or  minimum  value,  and,  at  this  instant,  all  the  electromotive  force 
is  used  to  overcome  the  resistance  of  the  circuit  since  L-dijdt  is 
zero.  The  time  /' ',  Fig.  14,  at  which  the  current  reaches  its 
maximum  value  is  later  than  the  time,  /,  at  which  the  electro- 
motive force  reaches  its  maximum  value.  In  some  cases,  how- 
ever, the  current  may  reach  its  maximum  value  before  the 
electromotive  force. 

The  curve  in  Fig.  15 
represents  the  electromo- 
tive force  of  an  alternator 
with  concentrated  arma- 
ture windings,  the  arma- 
ture core  being  of  the 
shape  shown  in  Fig.  10. 

25.  The  representation  of  al- 
ternating electromotive  force  and 
current  by  polar  coordinates. — 
The  successive  instantaneous  values 
of    an     alternating    electromotive 

force,  or  current,  may  be  represented  by  the  varying  lengths  of  a  rotating  radius 
vector,  elapsed  time  being  represented  by  the  angle  between  the  moving  radius 
vector  and  a  fixed  axis  of  reference.  Thus,  in  Fig.  16,  OE  represents  the  value  at  a 
given  instant  of  an  alternating  electromotive  force  and  07  the  value  at  the  same  in- 
stant of  an  alternating  current,  and  the  angle  0  represents  the  time  elapsed  since  a 
chosen  instant.  This  Fig.  16  represents  the  same  electromotive  force  and  current 
as  are  represented  by  Fig.  14. 

* 

26.  Instantaneous  and  average  power  delivered  by  an  alterna- 
tor.— Let  e  be  the  value  at  a  given  instant  of  the  electromotive 
force  of  an  alternator  and  i  the  value  of  the  current  at  the  same 


Fig.  16. 


28 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


instant.  Then  ei  is  the  power  in  watts  which  is  delivered  by  the 
alternator  at  the  given  instant,  and  the  average  value  of  ei  is  the 
average  power  delivered  by  the  alternator. 

Examples. — In  Fig.  17  the  full-line  curve  represents  the  elec- 
tromotive force  of  an  alternator  and  the  dotted  curve  represents 


Fig.  17. 

the  current  delivered  by  the  alternator  to  a  receiving  circuit  hav- 
ing inductance.  The  ordinates  of  the  dot-dash  curve  represent 
the  successive  instantaneous  values  of  the  power,  ei.  As  is 
shown  in  the  figure,  the  power  has  both  positive  and  negative 
values,  the  alternator  does  work  on  the  circuit  when  ei  is  posi- 
tive and  the  circuit  returns  power  to  the  alternator  when  ei  is 
negative,  and,  of  course,  while  ei  is  negative,  the  dynamo  is  mo- 
mentarily a  motor,  and  may  for  the  moment  return  power  to  the 
fly-wheel  of  the  engine. 

When  the  inductance  of  the  receiving  circuit  is  very  large,  the 
electromotive  force  and  current  curves  are  as  shown  in  Fig.  1 8, 


Fig.  18. 


the  instantaneous  power  ei  passes  through  approximately  simi- 
lar sets  of  positive  and  negative  values  as  shown  by  the  dot-and- 
dash  curve,  and  the  average  power  is  approximately  zero. 


THE   SIMPLE   ALTERNATOR.  29 

27.  Average  values  and  effective  values. — The  average  value 
of  an  alternating  current  or  electromotive  force  is  zero,  inasmuch 
as  similar  sets  of  positive  and  negative  values  occur.  The  aver- 
age value  of  an  electromotive  force  or  current  during  the  positive 
(or  negative)  part  of  a  cycle  is  usually  spoken  of  briefly  as  the 
average  of  mean  value,  and  is  not  zero. 

Effective  values.— Consider  an  alternating  current,  of  which  the 
instantaneous  value  is  i.  The  rate  at  which  heat  is  generated  in 
a  circuit  through  which  the  current  flows  is  Ri2,  where  R  is  the 
resistance  of  the  circuit,  and  the  average  rate  at  which  heat  is 
generated  in  the  circuit  is  R  multiplied  by  the  average  value  of  z2. 
A  continuous  current  which  would  produce  the  same  heating  ef- 
fect would  be  one  of  which  the  square  is  equal  to  the  average 
value  of  z2  or  of  which  the  actual  value  is  equal  to  ^average  i2. 
This  square  root  of  the  average  square  of  an  alternating  current 
is  called  the  effective  value  of  the  alternating  current.  Similarly 
the  square  root  of  the  average  square  of  an  alternating  electro- 
motive force  is  called  the  effective  value  of  the  alternating  elec- 
tromotive force. 

Ammeters  and  voltmeters  used  for  measuring  alternating  currents 
and  alternating  electromotive  forces  always  give  effective  values, 
and  in  specifying  an  alternating  electromotive  force  or  current  its 
effective  value  is  always  used. 

Example. — Consider  the  successive  instantaneous  values  (sepa- 
rated by  equal  time  intervals)  of  an  alternating  current.  The 
sum  of  these  values  divided  by  their  number  gives  their  average 
value.  Square  each  instantaneous  value.  Add  these  squares, 
divide  by  their  number  and  extract  the  square  root,  and  the  re- 
sult is  the  square-root-of-average-square,  or  effective,  value  of 
the  current. 

Form  factor  of  an  alternating  electromotive  force. — The  quotient, 
effective  value  of  an  alternating  electromotive  force,  divided  by 
the  average  value  of  the  electromotive  force  during  half  a  cycle, 
is  called  the/<?r;#  factor  of  the  electromotive  force,  inasmuch  as 
this  ratio  depends  upon  the  shape  of  the  electromotive  force  curve. 


30  ELEMENTS   OF   ALTERNATING   CURRENTS. 

In  the  case  of  a  sine-curve-electromotive-force  the  form  factor 
is  equal  to  1 .  1 1 ,  as  is  shown  in  Chapter  IV.  The  form  factor 
of  the  electromotive  force  represented  by  the  curve  in  Fig.  1 5  is 
greater  than  1 .  1 1 . 

28.  The  alternating  current  rectifier  is  an  arrangement  for  re- 
versing the  connections  of  a  receiving  circuit  with  each  reversal 
of  the  current  from  the  alternator,  so  that  the  current  may  flow 
always  in  the  same  direction  in  the  receiving  circuit.     The  recti- 
fier is  frequently  used  on  alternators  for  rectifying  the  main  cur- 
rent in  the  series  coils  on  the  field  for  the  purpose  of  providing 
increase  of  field  excitation  with  increase  of  current  output  of  the 
alternator.     In  this  case  the  rectifier  is  a  commutator  mounted 
on  the  armature  shaft.     This  commutator  has  as  many  bars  as 
there  are  poles  of  the  field  magnet  of  the  alternator.     These  bars 
are  wide  and  separated  by  quite  narrow  spaces  filled  with  mica. 
Let  these  bars  be  numbered  in  order  around  the  commutator. 
The  even-numbered  bars  are  connected  together  and  the  odd- 
numbered  bars   are  connected  together.     The   connecting  wire 
leading  from  one  terminal  of  the  alternator  armature  to  one  of 
the  collector  rings  is  cut  and  the  two  ends  thus  formed  are  con- 
nected, one  to  the  even-numbered  bars  of  the  rectifying  commu- 
tator and  the  other  to  the  odd-numbered  bars.     The  circuit  which 
is  to  receive  the  rectified  current  is  connected  to  two  brushes 
which  rub  on  the  rectifying  commutator,  these  brushes  being  so 
spaced  that  one  touches  an  odd-numbered  bar  when  the  other 
touches  an  even-numbered  bar.     These  brushes  are  carried  in  a 
rocker  arm,  which  is  moved  forwards  or  backwards  until  the 
brushes  are  passing  from  one  bar  to  the  next  at  the  instant  that 
the  alternating  current  from  the  alternator  is  passing  through 
the  value  zero.     The  proper  adjustment  of  the  brushes  is  indi- 
cated by  a  minimum  of  sparking. 

29.  The  fundamental  equation  of  the  alternator. — The  equation 
which  expresses  the  effective  value  of  the  electromotive  force  of 
an  alternator  in  terms  of  the  armature  speed  n,  the  number  of 


THE   SIMPLE  ALTERNATOR. 


31 


pairs  of  field  magnet  poles  /,  the  flux  <£  from  one  pole  of  the 
field  magnet,  and  the  total  number  of  armature  conductors  N 
which  cross  the  face  of  the  armature  is  called  the  fundamental 
equation  of  the  alternator.  This  equation  is  important  in  design- 
ing. It  is  derived  as  follows 
for  the  case  in  which  the  arma- 
ture conductors  are  concen- 
trated in  2p  slots,  one  to  each 
field  pole  as  shown  in  Fig.  19. 
Let  <1>  be  the  lines  of  flux 
from  one  pole,  then  one  arma- 
ture conductor  in  one  revolu- 
tion cuts  2/<l>  lines,  *  and  in 

one  second  it  cuts  2/4>#  lines,  which  is  the  average  electromotive 
force  (in  c.g.s.  units)  induced  in  one  armature  conductor.  We 
have,  therefore, 

Average  f  electromotive  force  of  alternator  in  volts 


I0 


The  ratio,  effective  electromotive  force  divided  by  average  electro- 
motive force,  is,  for  commercial  alternators,  approximately  equal 
to  i.n.J  Therefore,  the  effective  electromotive  force  of  an 
alternator  with  concentrated  armature  winding  is  approximately 


(19) 


or,  since  pn  is  the  frequency  according  to  equation  (17)  we  have 


*  Since  we  are  concerned  with  the  average  value  during  half  a  cycle  the  change  of 
sign  during  the  two  halves  of  a  cycle  is  to  be  ignored  and  the  flux  from  north  and 
from  south  poles  is  to  be  treated  without  regard  to  sign. 

fThat  is  the  average  during  half  a  cycle  as  explained  in  Article  27.  The  average 
during  a  whole  cycle  is  zero. 

J  See  Article  27. 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


in  which  <I>  is  the  magnetic  flux  from  one  pole,  and  N  the  total 
number  of  conductors  on  the  armature  which  are  connected  in 
series.  Sometimes  it  is  more  convenient  to  have  the  equation 
given  in  terms  of  armature  turns  instead  of  armature  conductors. 
The  formula  then  becomes 

4.44^77- 


io8 


(21) 


T  being  the  number  of  armature  turns  in  series  between  the  col- 
lector rings. 

30.  Experimental  determination  of  electromotive  force  curves. 
The  contact-maker. — A  disc  DD,  Figs.   20  (a)  and  20  (b),  fixed 


£1 


shaft 


ff 


© 


Fig.  20  (a). 

to  and  rotating  with  the  armature  shaft,  carries  a  pin  /,  which 
makes  momentary  electrical  contact,  once  per  revolution,  with  a 
jet  of  conducting  liquid  which  issues  from  a  nozzle  n.  This  noz- 
zle is  carried  on  a  pivoted  arm  a,  and  can  be  moved  at  will,  its 
position  being  read  off  the  divided  circle  cc.  One  terminal  of  an 


THE   SIMPLE   ALTERNATOR. 


33 


electrostatic  voltmeter  Q  is  connected  directly  to  one  brush  of 
the  alternator,  while  the  other  terminal  of  the  voltmeter  is  con- 
nected through  the  jet  and  pin  to  the  other  brush  of  the  alternator 
as  shown  in  Fig.  20  (a).  The  voltmeter  then  indicates  the  value 
of  the  electromotive  force  of  the  alternator  at  the  instant  of  con- 
tact of  jet  and  pin.  By  shifting  the  jet,  step  by  step,  around  the 


Fig.  20  (b). 

circle  successive  instantaneous  values  of  the  electromotive  force 
may  be  determined.  The  electromotive  force  passes  through  a 

complete  cycle  of  values  while  the  jet  is  shifted  -  of  a  revolution, 

/  being  the  number  of  pairs  of  poles  of  the  alternator.  In  order 
that  the  electromotive  force  acting  upon  the  electrostatic  volt- 
meter may  not  fall  off  appreciably  in  the  intervals  between  succes- 
sive contacts  of  pin  and  jet,  a  condenser  J  is  connected  as  shown 
in  Fig.  20  (a).  The  indications  of  an  electrostatic  voltmeter  are 
not  accurate  for  small  deflections  and  in  using  such  an  instru- 
ment for  measuring  a  comparatively  small  electromotive  force  a 
battery  of  known  electromotive  force  may  be  connected  in  the 
circuit  so  as  to  raise  the  electromotive  force  to  an  accurately 
measurable  value. 


34 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


In  the  determination  of  an  alternating  current  curve,  the  cur- 
rent is  sent  through  a  non-inductive  resistance  R,  Fig.  21,  and 

the  electromotive  force  between  the 
terminals  of  this  resistance  is  deter- 
mined as  before,  the  disc  DD  being 
fixed  to  the  armature  shaft  of  the 
alternator  which  is  furnishing  the  cur- 
rent. The  current  at  each  instant  is 
equal  to  the  electromotive  force  di- 
vided by  R. 

or 

brush  is  sometimes  used  instead  of 
the  liquid  jet  in  Fig.  20.  In  this  case 
the  pin  /  is  replaced  by  a  strip  of  metal  set  in  the  edge  of  a  cir- 
cular disc  of  hard  rubber  and  the  spring  rubs  continuously  upon 
the  edge  of  this  disc,  touching  the  metal  strip  momentarily  once 
per  revolution  of  the  disc. 


R 


•A/wvwH — '  uy  * 

i  I  Remark. — A  flat  metal  spring 

'I  hriish    is    sometimes    iispH    irmtpa< 


Fig.  21. 


PROBLEMS. 

15.  An  alternator  has  16  poles  and  its  speed  is  900  revolu- 
tions per  minute.     What  is  the  frequency  of  its  electromotive 
force  ?     What  is  the  duration  of  one  cycle  ?     Ans.  1 20  cycles 
per  second,  -j^-  second. 

1 6.  A  four-pole  alternator  makes  1,800  revolutions  per  minute. 
Each  pole  face  has  400  square  centimeters  of  area  and  spans 
54°  of  the  circumference  of  the  armature,  the  angle  between  the 
adjacent  pole  tips  being   36°.     The  armature  core  is  smooth, 
that  is,  not  slotted,  its  length  is  30  cm.,  and  the  field  intensity 
in  the  gap  space  is  6,000  units.     The  armature  is  wound  with 
four  wires,  one  per  field  pole.     These  wires  lie  on  the  face  of  the 
armature  parallel  to  the  axis  of  the  shaft,  they  are  spaced  90° 
apart,  and  they  are  connected  in  series  between  the  collecting 
rings.      Calculate  the  electromotive  force  induced  in  the  four  wires 
when  they  are  moving  under  the  pole  pieces.      Plot  the  electro- 


THE   SIMPLE   ALTERNATOR.  35 

motive  force  curve  of  the  alternator,  and  find  its  form  factor. 
Ans.  9.6  volts,  form  factor  1.29. 

17.  The  armature  winding  of  the  above  alternator  consists  of 
100  conductors  arranged  in  four  bands,  each   band  30°  wide. 
These  four  bands  replace  the  four  single  conductors  described  in 
problem  16,  and  all  100  conductors  are  in  series  between  the  col- 
lecting rings.     Calculate  the  electromotive  force  of  the  alterna- 
tor when  the  bands  of  conductors  are  wholly  under  the  pole 
pieces  ;   calculate  the  electromotive  force  when   the  bands   are 
half  under  the  pole  pieces.      Plot  the  electromotive  force  curve 
of  the  alternator  and  find  its  form  factor.     Speed  and  dimensions 
of  alternator  the  same  as  in  problem  16.     Ans.  240  volts,  120 
volts,  form  factor  1.165. 

1 8.  The  field  intensity  in  the  gap  space  in  problem  16  changes 
uniformly  from  4,500  units  under  the  leading  pole  tips  to  7,500 
units  under  the  trailing  pole  tips.      Plot  the  electromotive  force 
curve  of  the  four  conductor  winding  and  find  its  form  factor. 
Speed  and  dimensions  of  alternator  the  same  as  in  problem  1 6. 
Ans.  form  factor  =  1.38. 

IQ.  The  adjacent  field  pole  tips  of  an  alternator  touch  each 
other,  the  field  intensity  in  the  gap  space  is  uniform,  and  the 
armature  winding  consists  of  one  concentrated  bundle  of  conduc- 
tors for  each  field  pole.  What  is  the  form  factor  ?  Ans.  form 
factor  =  unity. 

20.  The  electromotive  force  of  an  alternator  passes  through  a 
complete  cycle  of  values,  while  the  magnetic  flux,  through  an 
armature  coil,  passes  through  a  cycle  of  values,  starting  out  from 
any  given  value  and  coming  back  to  that  value  again.     Show 
that  the  electromotive  force  of  an  inductor  alternator  of  the 
type  shown  in  Fig.  1 1   passes  through  2p  cycles  per  revolution 
of  the  set  of  inductors,  /  being  the  number  of  pairs  of  field 
magnet  poles,  NS,  NS,  etc. 

21.  An  alternator  has  8  poles,  and  its  speed  is  900  revolutions 
per  minute.     The  flux  from  one  pole  is  2,200,000  lines.     The 


36  ELEMENTS   OF   ALTERNATING   CURRENTS. 

armature  winding  is  concentrated  in  8  slots,  and  it  consists  of 
i  ,000  conductors,  all  of  which  are  connected  in  series.  What  is 
the  effective  electromotive  force  obtained  between  the  collector 
rings?  Assume  the  form  factor  to  have  the  value  1.16.  Ans. 
3,060  volts. 

22.  An  alternator  has  10  poles,  and  runs  at  a  speed  of  1,500 
revolutions  per  minute,  generating  2,000  volts.     The  flux  from 
one  pole  is  2,250,000  lines.      How  many  turns  must  there  be  on 
the  armature  if  they  are  all  connected  in  series  ?     Assume  the 
form  factor  to  be  1.16.     Ans.  306. 

23.  The  following  are  instantaneous  values,  in  volts,  of  the. 
electromotive  force  of  an  alternator,  taken  at  equal  intervals  dur- 
ing an  entire  cycle :  o,  30,  60,  80,  90,  95,  90,  80,  60,  30,  o,  —  30, 

—  60,  —  80,  —  90,  —  95,  —  90,  —  80,  —  60,  —  30  and  o.     The 
corresponding  values,  in    amperes,  of  the  current    are :  —  65, 
-  45,  -  25,  o,  25,  45,  65,  75,  78,  75,  65,  45,  25,  o,  -  25,  -  45, 

—  65,   —75,   —78,   —75  and   —65.     Find    the  instantaneous 
values  of  the  power,  plot  the  curves  of  electromotive  force,  of 
current  and  of  power,  and  find  the  average  power. 


CHAPTER   III. 

ALTERNATING   AMMETERS,   VOLTMETERS   AND 
WATTMETERS. 

31.  The  hot-wire  ammeter  and  voltmeter.* — In  these  instru- 
ments the  current  to  be  measured  is  sent  through  a  stretched 
wire.     The  wire,  heated  by  the  current,  lengthens  and  actuates 
a  pointer  which  plays  over  a  divided  scale. 

The  hot-wire  instrument ',  when  calibrated  by  continuous  currents, 
indicates  effective  values  of  alternating  currents,  and  when  cali- 
brated by  continuous  electromotive  forces  it  indicates  effective  values 
of  alternating  electromotive  forces. 

Proof.  — Consider  an  alternating  current  and  a  continuous  current  C  which  give  the 
same  reading.  These  currents  generate  heat  in  the  wire  at  the  same  average  rate. 
This  rate  is  RC*  for  the  continuous  current  and  R  X  average  i2  for  the  alternating 
current,  i  being  the  instantaneous  value  of  the  alternating  current.  Therefore  1?C2  — 
^X  average  i2  or  C2=  average  z2  or  C=  \/  average  i*.  Q-  E.  D. 

The  proof  for  electromotive  forces  is  similar  to  this  proof  for  currents. 

Remark. — The  readings  of  a  hot-wire  voltmeter  cannot  be  re- 
duced to  current  by  divjding  by  a  constant  factor  the  resistance 
of  the  instrument,  as  can  the  readings  of  most  other  types  of 
voltmeters,  inasmuch  as  the  resistance  of  the  instrument  varies 
greatly  with  the  changing  temperature  of  the  wire. 

32.  The  electro-dynamometer  used  as  an  ammeter. — The  essen- 
tial parts  of  the  electro -dynamometer  are  shown  in  Figs.  22  (a) 
and  22  (b).     These  figures  show  the  arrangement  of  the  parts  in 

*  All  voltmeters,  except  the  electrostatic  voltmeter,  are  essentially  ammeters.  That 
is,  the  electromotive  force  to  be  measured  produces  a  current  which  actuates  the  instru- 
ment. The  scale  over  which  the  pointer  plays  may  be  arranged  to  indicate  either  the 
value  of  the  current  or  the  value  of  the  electromotive  force. 

37 


ELEMENTS  OF  ALTERNATING  CURRENTS. 


Siemens'  type  of  instrument.  The  coil  A  is  held  stationary  by 
the  frame  of  the  instrument,  while  the  coil  B  is  mounted  at  right 
angles  to  A  and  is  hung  from  a  suspension.  This  movable  coil 
is  provided  with  flexible  or  mercury-cup  connections  aa,  and  the 
current  to  be  measured  is  sent  through  both  coils  in  series.  The 
force  action  between  the  coils  is  balanced  by  carefully  twisting 
a  helical  spring  ^,  one  end  of  which  is  attached  to  the  coil  B  and 
the  other  to  the  torsion  head  c.  The  observed  angle  of  twist 
necessary  to  bring  the  swinging  coil  to  its  zero  position  is  read 
off  by  means  of  the  pointer  d  and  the  graduated  scale  e.  The 

pointer  f  attached  to  the  coil 
shows  when  it  has  been  brought 
to  its  zero  position.  The  observed 
angle  of  twist  of  the  helical 
spring  affords  a  measure  of  the 
force  action  between  the  coils  and 
the  current  is  proportional  to 
the  square  root  of  this  angle  of 
twist.  In  other  forms  of  electro- 


Fig.  22  (a).  Fig.  22  (b). 

dynamometer  the  force  action  between  the  coils  moves  the  sus- 
pended coil  and  causes  the  attached  pointer  to  play  over  a 
divided  scale. 

The  electro-dynamometer  when  standardized  by  direct  currents 
indicates  effective  values  of  alternating  currents. 

Proof. — A  given  deflection  of  the  suspended  coil  depends  upon  a  definite  average 
or  constant  force  action  between  the  coils.     The  force  action  due  to  a  constant  cur- 


AMMETERS,  VOLTMETERS   AND   WATTMETERS.  39 

rent  c  is  he*  (proportional  to  t2),  and  the  average  force  action  due  to  an  alternating 
current  is  k  X  average  z2,  so  that  if  these  currents  gave  the  same  deflection,  we  have 
kc*  =  k  X  average  z2,  or  c2  =  average  i2,  or  c  =  V  average  i*.  Q.  E.  D. 

Remark.  —  The  electro-dynamometer  is  the  standard  instru- 
ment for  measuring  alternating  currents,  and  it  is  always  used  in 
accurate  measurements. 

33.  The  electro-dynamometer  used  as  a  voltmeter.  —  When  used 
as  a  voltmeter  the  coils  of  the  electro-dynamometer  are  made  of 
fine  wire,  and  an  auxiliary  non-inductive  resistance  is  usually 
connected  in  series  with  the  coils. 

When  the  inductance  of  the  electro-dynamometer  coils  is  small 
such  an  instrument,  when  calibrated  by  continuous  electromotive 
forces,  indicates  effective  values  of  alternating  electromotive  forces. 

When  it  is  certain  that  the  inductance  of  an  electro  -dynamom- 
eter is  negligibly  small  the  instrument  may  be  used  in  refined 
alternating  electromotive  force  measurements. 

Inductance  error  of  the  electro-dynamometer  used  as  a  voltmeter.  —  An  electro- 
dynamometer  which  has  been  calibrated  by  continuous  electromotive  force  indicates 
less  than  the  effective  value  of  an  alternating  electromotive  force.  The  following  dis- 
cussion of  this  error  for  the  case  of  harmonic  electromotive  force  presupposes  a 
knowledge  of  Chapters  IV.  and  V.  Let  JE£  be  the  reading  of  an  electro-dynamome- 
ter voltmeter,  when  an  alternating  electromotive  force  (harmonic),  of  which  the  ef- 
fective value  is  E,  is  connected  to  its  terminals,  that  is,  1$  is  the  continuous  electro- 
motive force,  which  gives  the  same  deflection  as  £,  and,  since  E  gives  the  same 
deflection  as  J£,  it  follows  that  the  effective  current  produced  by  E  is  equal  to  the 
continuous  current  produced  by  1£,  that  is, 


in  which  R  is  the  total  resistance  of  the  instrument,  L  its  inductance,  and  6>  =  27r/j 
where  f  is  the  frequency  of  the  alternating  electromotive  force.  Solving  equation 
(22)  for  E  we  have 

;  (23) 


that  is,  the  reading  of  the  instrument  must  be  multiplied  by  the  factor 


to  give  the  true  effective  value  of  an  harmonic  alternating  electromotive  force. 
Plunger  type  voltmeters  have  inductance  errors  also. 


40  ELEMENTS    OF   ALTERNATING   CURRENTS. 

34,  The  electrostatic  voltmeter. — Two  insulated  metal  plates, 
which  are  connected  to  the  terminals  of  a  battery,  or  to  any 
source  of  electromotive  force,  attract  each  other  with  a  force 
which  is  strictly  proportional  to  the  square  of  the  electromotive 
force.  This  principle  is  applied  in  the  electrostatic  voltmeter, 
which  consists  essentially  of  a  fixed  plate  and  a  suspended  plate 
to  which  a  pointer  is  attached.  The  terminals  of  the  electromo- 
tive force  to  be  measured  are  connected  to  these  plates, 

Such  an  instrument,  when  calibrated  by  continuous  electromotive 
force,  indicates  effective  values  of  alternating  electromotive  force. 

Proof. — A  given  deflection  of  the  suspended  plate  depends  upon  a  definite  average 
or  constant  force  action  between  the  plates.  The  force  action  due  to  a  constant  elec- 
tromotive force  1£  is  KJ§*  (proportional  to  J^2),  and  the  average  force  action  due  to 
an  alternating  electromotive  force  e  is  A"X  average  e2.  If  these  electromotive  forces 
give  equal  deflections  the  force  KJ$2  is  equal  to  the  average  force  Ky^  average  e2  so 
that  J£2  =  average  e2,  or  1$  =  V average  e2.  Q.  E.  D. 

The  electrostatic  voltmeter  is  the  standard  instrument  for 
measuring  alternating  electromotive  forces,  especially  for  the 
measurement  of  very  high  electromotive  force.  Further,  with 
high  electromotive  forces  the  electrostatic  attraction  of  parallel 

metal  plates  is  great 
enough  to  be  accu- 
rately measured  by  a 
balance  and  in  this  case 
the  electromotive  force 

A          I     a     \         A  between      the      plates 

air  air  , 

— (constant  electromotive 

force,  or  effective  value 

Fig.  23.  .  .  , 

of  an  alternating  elec- 
tromotive force)  may  be  calculated  independently  of  calibra- 
tions of  any  kind.  An  instrument  arranged  for  the  absolute* 
measurement  of  electromotive  force  in  this  way  is  called  an  abso- 
lute electrometer. 

The  absolute  electrometer  consists  of  two  parallel  metal  plates, 
AaA  and  BB,  Fig.  23.  The  central  portion  a  of  the  upper 

*That  is,  the  measurement  in  terms  of  mechanical  units  offeree,  distance,  etc. 


AMMETERS,  VOLTMETERS   AND   WATTMETERS.  41 

plate,  while  remaining  in  electrical  communication  with  AA,  is 
detached  and  suspended  from  one  arm  of  a  balance  beam  as 
shown.  The  electromotive  force  E  between  AaA  and  BB,  in 
volts,  is  given  by  the  formula 


in  which  /MS  the  observed  downward  pull  on  a  in  dynes,  d  is  the 
distance  apart  of  the  plates  in  centimeters  and  a  is  the  area  in 
cm2  of  the  detached  portion  a. 

35.  The  spark  gauge.  —  The  high  electromotive  forces  used  in 
break-down  tests  are  usually  measured  by  means  of  the  spark 
gauge.     This  consists  of  an  adjustable  air  gap  which  is  adjusted 
until  the  electromotive  force  to  be  measured  is  just  able  to  strike 
across  in  the  form  of  a  spark.     The  electromotive  force  is  then 
taken  from  empirical  tables  based  upon  previous  measurements 
of  the    electromotive    force    required    to    strike    across  various 
widths  of  gap.     In  the  spark  gauge  of  the  General  Electric  Co. 
the  spark  gap  is  between  metal  points,  one  of  which  is  attached 
to  a  micrometer  screw  by  means  of  which  the  gap  space  may 
be  adjusted  and  measured.     The  striking  distance  in  any  spark 
gauge  varies  greatly  with   the  condition  of  the  points.      It  is, 
therefore,  necessary  to  see  that  the  points  are  well  polished  be- 
fore taking  measurements. 

36.  Plunger  type  ammeters  and  voltmeters.  —  In  instruments  of 
this  type  the  current  to  be  measured  passes  through  a  coil  of 
wire  which  magnetizes  and  attracts  a  movable  piece  of  soft  iron 
to  which  the  pointer  is  fixed. 

A  plunger  meter  (ammeter  or  voltmeter)  should  be  calibrated 
under  the  conditions  in  which  it  is  to  be  used.  Thus,  if  a  plunger 
instrument  is  to  be  used  as  an  ammeter  for  alternating  currents 
of  a  given  frequency  it  should  be  calibrated  by  currents  of  this 
frequency,  these  currents  being,  for  the  purpose  of  the  calibra- 
tion, measured  by  a  standard  alternating  current  ammeter,  such 


42 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


as  an  electrodynamometer.  The  indications  of  a  plunger  instru- 
ment do  not,  however,  vary  greatly  with  frequency  and  such  in- 
struments are  used  for  approximate  measurements  without  regard 
to  frequency. 

The  Thomson  inclined  coil  meter  of  the  General  Electric  Co.  is 
of  the  plunger  type.  The  essential  parts  of  this  instrument  are 
shown  in  Fig.  24.  A  coil  A,  through  which  flows  the  current 


Fig.  24- 

to  be  measured,  is  mounted  with  its  axis  inclined  as  shown.  A 
vertical  spindle  mounted  in  jeweled  bearings  and  controlled  by  a 
hair-spring  passes  through  the  coil,  and  to  this  spindle  are  fixed 
a  pointer  b  and  a  vane  of  thin  sheet-iron  a.  This  vane  of  iron 
is  mounted  obliquely  to  the  spindle.  When  the  pointer  is  at  the 
zero  point  of  the  scale  the  iron  vane  a  lies  nearly  across  the  axis 
of  the  coil,  and  when  a  current  passes  through  the  coil  the  vane 
tends  to  turn  until  it  is  parallel  to  the  axis  of  the  coil,  thus  turn- 
ing the  spindle  and  moving  the  attached  pointer  over  the  cali- 
brated scale. 

37,  The  potential  method  for  measuring  alternating-  current. — 
The  alternating  current  to  be  measured  is  passed  through  a 
known  non-inductive  resistance  R  and  the  electromotive  force 
between  the  terminals  of  this  resistance  is  measured  by  a  volt- 
meter. The  current  (effective  value)  is  then  equal  to  the  electro- 
motive force  (effective  value)  divided  by  the  resistance. 

The  calorimetric  method  for  measuring  alternating  current. — 
The  current  to  be  measured  is  passed  through  a  known  resist- 


AMMETERS,  VOLTMETERS   AND   WATTMETERS. 


43 


ance  which  is  submerged  in  a  calorimeter  by  means  of  which  the 
heat  H  which  is  generated  in  the  resistance  in  an  observed  in- 
terval of  time  /  is  determined.  This  heat  being  expressed  in 
joules  we  have 


//=  PRt 
in  which  /  is  the  effective  value  of  the  current. 


(25) 


MEASUREMENT  OF  POWER  IN  ALTERNATING  CIRCUITS.* 
38.  The  three-voltmeter  method. — A  non-inductive  resistance 
R,  Fig.  25,  is  connected  in  series  with  the 
circuit  be  in  which  the  power  P,  to  be  deter- 
mined, is  expended.  The  electromotive  forces 
Ev  between  ab,  Ez  between  be  and  £3  between 
ac,  are  observed  by  means  of  a  voltmeter  as 


main 


main 


nearly  simultaneously  as  possible.     Then 


2R 


(26) 


Fig.  25. 


Proof.  —  Let  ev  ez  and  e3,  be  the  instantaneous  electro- 
motive  forces  between  ab,  be  and  ac,  respectively,  then 


Average  <?32  =  average  e^  -\-  2  average  e^  -j-  average  <?22  f 


but  E£  =  average  e3*t  £lt  =  average  e^  and  Ef  =  average  /-22.  Further  -  *  is  the  in- 
stantaneous current  in  abcy  -±  -  e.2  is  the  instantaneous  power  expended  in  be  and  aver- 
age f  jjt'^j)  or  T>  X  average  (^2)  is  the  average  power  P  expended  in  be  so  that 


average  (e^t)  =  RP.     Therefore  equation  (iii)  becomes 

or 

p=F* 


(iv) 


2R 


Q.  E.  D. 


*In  alternating  circuits  power  cannot  be  measured  by  means  of  an  ammeter  and  a 
voltmeter  as  in  the  case  of  direct  current  for  the  reason  that  the  power  expended  is  in 
general  less  than  the  product  of  effective  electromotive  force  into  effective  current  on 
account  of  the  difference  in  phase  of  the  current  and  electromotive  force. 

f  For  proof  of  (iii)  see  proposition,  Article  49. 


44 


ELEMENTS  OF  ALTERNATING  CURRENTS. 


39.  The  three-ammeter  method  for  measuring  power. — The  cir- 
cuit CC,  Fig.  26,  in  which  the  power  P  to  be  measured  is  ex- 
pended, is  connected  in  parallel  with 
a  non-inductive  resistance  R,  and 
three  ammeters  are  placed  as  shown. 
Then 

/>-§(^#-#)      (27) 

in  which  fv  I2  and  /3  are  the  currents 
indicated  by  the  three  ammeters. 

Proof. — Let  1\,  iz  and  z"3  be  the  instanta 
neous  values  of  the  currents  flt  /2  and  Ss.    Then 


or 
or 
But 


Fig.  26.  73  =  'i       l!l 

'7  =  '\2  +  2t\t'2  +  >** 

average  ?32=  average  z'j2  -j-  2  average  (?a/2)  4  average  ?'22 
!2  =  average  z^2,     /22  =  average  z'22     and  //  =  average  ;32. 


C") 

(iii) 


Further,  the  instantaneous  electromotive  force  between  the  terminals  of  R  or  of  CCis 
Riv  so  that  Rizi^  is  the  instantaneous  power  expended  in  CC,  and  R^  average  (z'jZ°2) 

p 

is  the  average  power  P  expended  in  CC.     Therefore,  average  (z'j^)  =~  and  equa- 
tion (iii)  becomes 

or 

P=-(f2—f2_f2\  Q.   E.   D. 

2 

Combination  method. — The  three-ammeter  method  for  measur- 
ing power  may  be  modified  by  using  the  potential  method  for 
measuring  /2,  Fig.  26.  In  this  case  the  electromotive  force  be- 
tween the  terminals  of  R,  Fig.  26,  is  measured  by  means  of  a 

voltmeter,  so  that  /2  ==  -~  where  E2  is  the  voltmeter  reading. 


40.  The  wattmeter. — The  wattmeter  is  an  electrodynamom- 
eter,  of  which  one  coil  a,  Fig.  27,  made  of  fine  wire,  is  connected 
to  the  terminals  of  the  circuit  CC,  in  which  the  power  to  be 


AMMETERS,  VOLTMETERS   AND   WATTMETERS.          45 

measured  is  expended.     The  other  coil  b,  made  of  large  wire,  is 
connected  in  series  with  CC,  as  shown. 

The  fine  wire   coil  a  is  movable,  and  Jnaisi 

carries    the    pointer    which    indicates 
the  watts  expended  in  CC. 

Such  an  instrument  when  calibrated 
with  continuous  current  and  electromo- 
tive force  indicates  power  accurately 
when  used  with  alternating  currents, 
presided  the  inductance  of  the  circuit  ain 


ar  is  small*  Fig-27- 

Proof. — A  given  deflection  of  the  movable  coil  a  depends  upon  a  certain  average  or 
constant  force  action  between  the  coils  Consider  a  continuous  electromotive  force  1J 

T? 

which  produces  a  current  —  in  a  and  a  current  C  in  CC  and  b.     The  force  action  be- 
tween the  coils  is  proportional  to  the  product  of  the  currents  in  a  and  l>,  that  is,  the 
-c\ 

force  action  is  k  . C,  where  k  is  a  constant. 

r 

Consider  an  alternating  electromotive  force  of  which  the  instantaneous  value  is  e  : 
this  produces  a  current  —  through  a  (provided  the  inductance  of  a  is  zero)  and  a  cur- 
rent i  in  CC  and  b.  The  instantaneous  force  action  between  thecoils  is  k  .  —  .  i  and 

the  average  force  action  is  —  .  average  («').     If  this  alternating  electromotive  force 
gives  the  same  deflection  as  the  continuous  electromotive  force  then 

—  X  average  (ei) '  —  I$C  — 

or  average  (ei)  =  I$C. 

That  is,  the  given  deflection  indicates  the  same  power  whether  the  currents  are 
alternating  or  direct.  Q.  E.  D. 

Remark. — A  good  wattmeter  is  the  standard  instrument  for 
measuring  power  in  alternating-current  circuits.  The  three -am- 
meter and  the  three-voltmeter  methods  are  troublesome  and 
slight  errors  of  observation  may  in  some  cases  lead  to  very  great 
errors  in  the  result. 


The  compensated  wattmeter  of  the  Weston  Electrical  Instrument  Co. — In  the  above 

*  Small,  that  is,  in  comparison  with  — -  ;  where  r  is  th< 
r,  Fig.  27,  and/is  the  frequency  of  the  alternating  current. 


Small,  that  is,  in  comparison  with  — -  ;  where  r  is  the  total  resistance  of  a  and 


46 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


proof  the  current  flowing  through  a  and  r,  Fig.  27,  is  assumed  to  be  negligible  in  com- 
parison with  the  current  in  CC,  so  that  the  current  in  the  coil  b  is  sensibly  the  same  as 
the  current  in  CC.  This  is  not,  however,  always  the  case.  In  fact  the  instrument 
indicates  the  total  power  expended  in  a,  in  r,  and  in  CC.  Let  C  be  the  current  in 
CC  and  let  a  be  the  current  in  a  and  r.  Then  the  current  in  b  is  C-\-  a,  and  the  force 
action  upon  the  movable  coil  is  proportional  to  the  product  a(  C-\-  a),  instead  of  being 
proportional  to  the  product  aC. 

In  the  compensated  wattmeter  of  the  Weston  Co.  the  wire  leading  over  to  the  coil 
a,  connected  as  shown,  is  laid  alongside  of  each  and  every  turn  of  wire  in  coil  b. 
Then  current  C  -\-  a  flows  down  through  b,  current  a  flows  back  alongside  of  the  wire 
of  coil  b,  and  the  result  is  the  same  as  if  the  current  a  were  subtracted  from  the  current 
C-\-  a  so  far  as  the  magnetic  action  of  the  coil  b  is  concerned. 

41.  The  recording  wattmeter  is  an  instrument  for  summing  up 
the  total  work  or  energy  expended  in  a  circuit. 

The  Thomson  recording 
wattmeter  is  a  small  electric 
motor  without  iron,  the 
field  and  armature  coils  of 
which  constitute  an  elec- 
trodynamometer.  The  field 
coils  BB  of  this  motor, 
Fig.  28,  are  connected  in 
series  with  the  circuit  CC 
in  which  the  work  to  be 
measured  is  expended. 

The    armature    A,    to- 
gether   with    an   auxiliary 
Fig- 28>  non-inductive  resistance  R, 

is  connected  between  the  terminals  of  the  circuit  CC,  as  shown. 
Current  is  led  into  the  armature  by  means  of  the  brushes  dd 
pressing  on  a  small  silver  commutator  e. 

Discussion  of  the  Thomson  recording  wattmeter. — The  driving  torque,  acting  upon 
the  armature,  is  proportional  to  the  rate  at  which  work  is  spent  in  the  circuit  CC(i.  <?., 
to  the  power  expended,  as  explained  in  Article  41 ).  The  instrument  is  so  constructed 
that  the  speed  of  the  armature  is  proportional  to  this  driving  torque  or  to  the  power 
spent  in  CC.  That  is,  the  rate  of  turning  of  the  armature  is  proportional  to  the  rate 
at  which  work  is  done  in  the  circuit  CC,  so  that  the  total  number  of  revolutions  turned 
by  the  armature  is  proportional  to  the  total  work  expended  in  the  circuit  CC. 

To  make  the  armature  speed  proportional  to  the  driving  torque  the  armature  is 


AMMETERS,  VOLTMETERS    AND   WATTMETERS.          47 

mounted  so  as  to  be  as  nearly  as  possible  free  from  ordinary  friction  and  a  copper  disk 
f,  Fig.  28,  is  mounted  on  the  armature  spindle  so  as  to  rotate  between  the  poles  of 
permanent  steel  magnets  MM.  To  drive  such  a  disk  requires  a  driving  torque  pro- 
portional to  its  speed. 

The  starting  coil. — In  the  above  discussion  it  is  assumed  that 
the  torque  which  opposes  the  motion  of  the  armature  A,  Fig.  28, 
is  proportional  to  the  speed  of  the  armature.  In  fact,  however, 
this  opposing  torque  may  be  considered  in  two  parts  :  1st,  the 
torque  to  overcome  friction,  and  2d,  the  torque  required  to  over- 
come the  damping  action  of  the  magnets  on  the  copper  disk. 
The  first  part  of  the  torque  may  be  taken  to  be  approximately 
constant,  while  the  second  part  is  accurately  proportional  to  the 
speed.  Therefore  an  arrangement  for  exerting  on  the  armature 
a  constant  torque,  sufficient  to  overcome  friction,  would  largely 
eliminate  errors  due  to  friction.  This  is  accomplished  in  the 
Thomson  meter  by  supplementing  the  field  coils  B,  Fig.  28, 
with  an  auxiliary  field  coil  connected  in  the  armature  circuit. 
This  auxiliary  field  coil  is  called  a  starting  coil.  So  long  as  the 
electromotive  force  between  the  mains  does  not  vary,  the  current 
in  the  starting  coil  is  constant  and  it,  therefore,  exerts  a  constant 
torque  upon  the  armature.  If,  however,  the  electromotive  force 
between  the  mains  varies,  the  torque  due  to  the  starting  coil 
varies  with  the  square  of  the  electromotive  force. 

Remark. — The  induction  wattmeter  is  a  sort  of  induction  motor. 
It  is  described  in  Chapter  XIV. 


PROBLEMS. 

24.  The  spring  of  a  Siemens  electrodynamometer  is  twisted 
through  an  angle  of  220°  to  balance  the  force  action  of  a  current 
of  1 8. 8  amperes.     What  current  will  require  a  twist  of  165°? 
Ans.  9.35  amperes. 

25.  The  angle  of  twist  of  a  Siemens  electrodynamometer  can 
be  read  to  ^  of  a  degree.     What  are  the  relative  errors  in  cur- 
rent due  to  an  error  of  ^  of  a  degree  when  the  total  twist  is  10° 


48  ELEMENTS   OF   ALTERNATING   CURRENTS. 

and  when  the  total  twist  is  100°  ?  Ans.  The  error,  in  amperes, 
is  3.16  times  as  great  in  case  of  the  smaller  deflection,  and  the 
percentage  error  is  ten  times  as  great. 

26.  The  electromotive  force  e  which  produces  a  deflection  d 
scale  divisions  of  an  electrostatic  voltmeter  is  approximately  : 
e  =  kVd.     What  are  the  relative  errors  in  electromotive  force 
due  to  an  error   of  -^   oif  a  division   in  the  reading  when  the 
total  deflection  is  10  divisions,  and  when  the  total  deflection  is 
100  divisions  ?     Ans.  10  :  I. 

27.  The  fine  wire  coil  of  a  wattmeter  has  500  ohms  resistance. 
The  wattmeter  indicates   62  watts  when  used    to  measure  the 
power  delivered  to  a  I  lo-volt  lamp,  the  fine  wire  coil  being  con- 
nected to  the  terminals  of  the  lamp.     What  is  the  true  power 
delivered  to  the  lamp?     Ans.  38  watts. 

28.  When  a  hot-wire  voltmeter  indicates   50  volts  a  current 
of  0.05  ampere  flows  through  the  instrument.     When  the  same 
voltmeter  indicates  100  volts,  0.07  ampere  flows  through  the  in- 
strument.    What  is   the   resistance  of  the   instrument  in   each 
case?     Ans.  1,000  ohms,  1,428.6  ohms. 

29.  A  Thomson  wattmeter  without  a  starting  coil  starts  at 
75  watts  load.     The  wattmeter  is  adjusted  to  give  true  record 
when  run  at  a  500- watt  load.     What  will  the  instrument  indicate 
when  run  on  a  constant  load  at  200  watts  for  4  hours,  running 
friction  being  assumed  to  be  equal  to  half  the  starting  friction  ? 
Ans.  702.6  watthours. 

30.  The  above  wattmeter  is  provided  with  a  starting  coil  so  as 
to  start  with  a  4O-watt  load  on  iio-volt  mains.     At  what  load 
will  the  instrument  start  on  5 5 -volt  mains?     Ans.  66.25  watts. 

31.  The  above  wattmeter  with  its  starting  coil  is  adjusted  to 
read  correctly  at  a  load  of  500  watts  on  I  lo-volt  mains.     At  what 
load  will  it  read  correctly  on  5  5 -volt  mains  ?     Ans.  5,750  watts. 

Suggestion. — Let  x  be  required  watts.  The  total  driving 
torque  in  first  case  minus  running  friction  is  to  be  to  total  driv- 
ing torque  in  second  case  minus  running  friction  as  500  is  to  x. 


CHAPTER   IV. 


HARMONIC  ELECTROMOTIVE  FORCE  AND  CURRENT. 

42.  Definition  of  harmonic  electromotive  force  and  current. — A 
line  OP,  Fig.  29,  rotates  at  a  uniform  rate,  /  revolutions  per  sec- 
ond, about  a  point  0,  in  the  direction  of 
the  arrow  gh.  Consider  the  projection 
Ob  of  this  rotating  line  upon  the  fixed 
line  AB,  this  projection  being  considered 
positive  when  above  0  and  negative  when 
below  0.  An  Jtarmonic  electromotive  force 
(or  current)  is  an  electromotive  force  (or 
current)  which  is  at  each  instant  propor- 
tional to  the  line  Ob,  Fig.  29.  The  line 
Ob  represents  at  each  instant  the  actual 
value  e  of  the  harmonic  electromotive 
force  to  a  definite  scale,  and  the  length 
of  the  line  OP  (which  is  the  maximum  length  of  Ob)  represents 
the  maximum  value  T&  of  the  harmonic  electromotive  force  to  the 
same  scale.  The  line  Ob  passes  through  a  complete  cycle  of 
values  during  one  revolution  of  OP,  and  so  also  does  the  har- 
monic electromotive  force  e.  Therefore  the  revo- 
lutions per  second  /  of  the  line  OP  is  the  fre- 
quency of  the  harmonic  electromotive  force  e. 
The  rotating  lines  E  and  J,  Fig.  30,  of  which 
the  projections  on  a  fixed  line  (not  shown  in  the 
figure)  represent  the  actual  instantaneous  values 
e  and  i  of  an  harmonic  electromotive  force  and  an  harmonic 
current  are  said  to  represent  the  harmonic  electromotive  force  and 

49 


Fig.  29. 


50      ELEMENTS  OF  ALTERNATING  CURRENTS. 

current  respectively.  Of  course,  the  rotation  of  the  lines  E  and 
I  is  a  thing  merely  to  be  imagined. 

43.  Algebraic  expression  of  harmonic  electromotive  force  and 
current, — The  line  OP,  Fig.  29,  makes  f  revolutions  per  second, 
and,  therefore,  it  turns  through  27rf  radians  per  second,  since 
there  are  27T  radians  in  a  revolution,  that  is 

&>  =  27rf  (28) 

in  which  co  is  the  angular  velocity  of  the  line  OP  in  radians  per 
second.  Let  time  be  reckoned  from  the  instant  that  OP  coin- 
cides with  Oa,  then  after  /  seconds  OP  will  have  turned  through 
the  angle  ft  —  cot,  and  from  Fig.  29  we  have 

Ob  =  OP  sin  £  =  OP  sin  tot 

But  Ob  represents  the  actual  value  e  of  the  harmonic  electromo- 
tive force  at  the  time  t  and  OP  represents  its  maximum  value  E, 
therefore 

e  =  E  sin  cot  (29) 

is  an  algebraic  expression  for  the  actual  value  e  of  an  harmonic 
electromotive  force  at  time  /,  E  being  the  maximum  value  of 

CO 

e,  and  —  being  the  frequency  according  to  equation  (28). 

Similarly 

z  =  JTsin  cot  /^o) 

is  an  algebraic  expression  for  the  actual  value  i  of  an  harmonic 
current  at  time  t,  I  being  the  maximum  value  of  i. 

Remark  i. — If  time  is  reckoned  from  the  instant  that  OP, 
Fig.  29,  coincides  with  the  line  Ob,  then  equations  (29)  and  (30) 
become 

e  =  E  cos  cot 

i=  I  cos  &)/ 

Remark  2. — The  curve  which  represents  an  harmonic  electro- 
motive force  or  an  harmonic  current  (see  Article  24)  is  a  curve 
of  sines. 


HARMONIC    ELECTROMOTIVE   FORCE   AND    CURRENT.    51 

Remark  3. — A  great  many  alternators,  especially  those  with 
distributed  armature  windings,  generate  electromotive  forces 
which  are  very  nearly  harmonic.  Calculations  in  connection 
with  the  design  of  alternating  current  apparatus  are  simple 
enough  to  be  practicable  only  when  the  electromotive  forces  and 
currents  are  assumed  to  be  harmonic.  Hereafter,  then,  when 
speaking  of  alternating  electromotive  forces  and  currents,  it  will 
be  understood  that  they  are  harmonic,  unless  the  contrary  is 
expressly  stated. 

44.  Definitions.*    Cycle. — A  cycle  is  one  complete  set  of  values 
(positive  and  negative)  through  which  an  electromotive  force  or 
current  repeatedly  passes.     The  frequency  is  the  number  of  cycles 
passed  through  per  second.     The  period  is  the  duration  of  one 
cycle.      For  example,  an  alternator  generates  electromotive  force 
at  a  frequency  of  60  cycles  per  second  ;  the  period  is  ^  of  a 
second  and  the  angular  velocity  of  the  line  OP,  Fig.  29,  is  60 
revolutions  per  second  or  377  radians  per  second  (  =  &>  ). 

Synchronism. — Two  alternating  electromotive  forces  or  currents 
are  said  to  be  in  synchronism  when  they  have  the  same  frequency. 
Two  alternators .  are  said  to  run  in  synchronism  when  their 
electromotive  forces  are  in  synchronism.  , 

45.  Phase  difference. — Consider   two  synchronous    harmonic 
electromotive  forces  el  and  ev     Suppose  that  ev  passes  through 
its  maximum  value  before  e2 ;  then  el  and  e2  are  said  to  differ  in 


*  The  definitions  of  cycle  and  frequency  given  in  Article  23  are  here  repeated  for 
the  sake  of  clearness.  All  definitions  given  in  this  article  apply  to  alternating  cur- 
rents and  electromotive  forces  of  any  character  as  well  as  to  harmonic  electromotive 
forces  and  currents. 


ELEMENTS  OF  ALTERNATING  CURRENTS. 


phase.  The  line  1^,  Fig.  31,  which  represents  el  must  be  ahead 
of  the  line  B2  which  represents  e^  as  shown  in  the  figure.  The 
angle  6  is  called  the  phase  difference  of  e^  and  e2 .  The  lines  l$l 
and  E2  are  supposed  to  be  rotating  about  0  in  a  counter-clock- 
wise direction  as  explained  in  Article  42. 

When  the  angle  6,  Fig.  31,  is  zero,  as  shown  in  Fig.  32,  the 
electromotive  forces  el  and  e2  are  said  to  be  in  phase.  In  this 
case  the  electromotive  forces  increase  together  and  decrease 
together ;  that  is,  when  el  is  zero  ez  is  also  zero,  when  el  is  at  its 
maximum  value  so  also  is  ^9,  etc. 

When  6  =  90°,  as  shown  in   Fig.   33,  the  two  electromotive 
forces  are  said  to  be  in  quadrature.     In  this  case  one  electromo- 
tive force  is  zero  when  the  other  is  a  maxi- 
mum, etc. 

When  0  =  180°,  as  shown  in  Fig.   34,  the 
two  electromotive  forces  are  said  to  be  in  op 


Fig.  33.  Fig.  34. 

position.  In  this  case  the  two  electromotive  forces  are  at  each 
instant  opposite  in  sign  and  when  one  is  at  its  positive  maximum 
the  other  is  at  its  negative  maximum,  etc. 

46.  Composition  and  resolution 
of  harmonic  electromotive  forces 
and  currents,  (a)  Composition. — 
Consider  two  synchronous  har- 
monic electromotive  forces  el  and 
e2  represented  by  the  lines  JSt 
Fig'  35>  and  E2,  Fig.  35.  The  sum  el  -f 

ez  is  an  harmonic  electromotive  force  of  the  same  frequency  and 
it  is  represented  by  the  line  JJ.  This  is  evident  when  we  consider 


HARMONIC    ELECTROMOTIVE   FORCE   AND    CURRENT.    53 


that  the  projection,  on  any  line,  of  the  diagonal  of  a  parallelogram 
is  equal  to  the  sum  of  the  projections  of  the  sides  of  the  parallel- 
ogram. 

Corollary. — The  sum  of  any  number  of  synchronous  electro- 
motive forces  (or  currents)  is  another  electromotive  force  (or  cur- 
rent) of  the  same  frequency  which  is  represented  in  phase  and 
magnitude  by  the  line  which  is  the  vector  sum  of  the  lines  which 
represent  the  given  electromotive  forces  (or  currents).  Thus  the 
lines  B^ ,  E2  and  I£3 ,  Fig.  36,  represent  three 
given  synchronous  harmonic  electromotive 
forces  and  the  line  B  (the  vector  sum  of 
Uj ,  J$2  and  J53)  represents  an  harmonic  elec- 
tromotive force  which  is  the  sum  of  the  given 
electromotive  forces. 

(b)  Resolution. — A  given  harmonic  elec- 
tromotive force  (or  current)  may  be  broken 
up  into  a  number  of  harmonic  parts  of  the 
same  frequency  by  reversing  the  process  of 
composition.  For  example,  the  line  B,  Fig. 
36,  represents  a  given  harmonic  electromotive  force  which  may  be 
split  up  into  the  three  electromotive  forces  represented  by  the 


Fig.  36. 


lines 


and  IJ 


47.  Examples  of  composition  and  reso- 
lution. 

(a)  Two  alternators  A  and  B  running 


Fig.  37. 


in  synchronism  are  connected  in  series  between  the  mains  as 
shown  in  Fig.  37.  If  the  electromotive  forces  of  A  and  B  are  in 
phase  the  electromotive  force  between  the  mains  will  be  simply 
the  numerical  sum  of  the  electromotive  forces  of  A  and  B.  If, 


54 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


jnain 


Fig.  39. 


on  the  other  hand,  the  electromotive  forces  of  A  and  B  differ  in 
phase  the  state  of  affairs  will  be  as  represented  in  Fig.  38  ;  in 
which  the  lines  A  and  B  represent  the  electromotive  forces  of  the 

alternators  A  and  B  respec- 
tively, 6  is  the  phase  differ- 
ence of  A  and  B,  and  the 
line  JJ  represents  the  elec- 
tromotive force  between  the 
mains. 

(ft)  Two  alternators  A  and 
B  running  in  synchronism 
are  connected  in  parallel 
between  the  mains  as  shown  in  Fig.  39.  Let  the  lines  A  and 
B,  Fig.  40,  represent  the  currents  given  by  A  and  B  respectively, 
the  phase  difference  being  0; 
then  the  current  in  the  main 
line  is  represented  by  I. 

(c)  Two  circuits  A  and  B  are 
connected  in  series  between  the 
mains  of  an  alternator  as  shown  Fig>  40> 

in  Fig.  41.     The  line  U,   Fig.  42,  represents  the  electromotive 
force  between  the  mains,  the  line  A  represents  the  electromotive 
.  force  between  the   terminals  of  the 

circuit  A,  and  the  line  B  represents 
the  electromotive  force  between  the 
terminals  of  the  circuit  B.  The  cir- 


Fig.  41. 


cuits  A  and  B  are  supposed  to  have  inductance.  If  one  of  the 
circuits  A  or  B  contains  a  condenser,  then  the  electromotive  forces 
A  and  B,  Fig.  42,  may  be  nearly  opposite  to  each  other  in  phase. 


HARMONIC    ELECTROMOTIVE   FORCE   AND    CURRENT.    55 


and  A  and  B  may  each  be  indefinitely  greater  than  the  electro- 
motive force  E  between  the  mains. 

(d)  Two  circuits  A  and  B,  Fig.  43,  are  connected  in  parallel 
across  the  terminals  of  an  alternator  as  shown.     The  current  / 

from  the  alternator  is  related  to  the 
currents  A  and  B  as  shown  in  Fig.  44. 
If  one  of  the  circuits  A  or  B  contains 


main, 


Fig.  43. 


B 

Fig.  44. 


a  condenser,  then  the  currents  A  and  B  may  be  nearly  opposite 
to  each  other  in  phase  and  the  currents  A  and  B  may  each  be 
indefinitely  greater  than  the  current  /  from  the  alternator. 

48.  Rate  of  change  of  harmonic  electromotive  forces  and  cur- 
rents.— Consider  the  harmonic  current  [see  equation  (30)] : 

i  =  I  sin  at  (a) 

When  this  current  is  sent  through  an  inductive  circuit  an  electro- 
motive force  L  -j-  is  at  each  instant  required  to  make  the  current 
dt 

increase  or  decrease.  In  the  study  of  alternating  currents  in  in- 
ductive circuits  it  is,  therefore,  necessary  to  consider  the  rate  of 

di 
change  -j-  of  the  current. 

Differentiating  the  above  expression  for  i  with  respect  to  time 
we  have 


or 


di 

-j-  =  O) I  COS  O)t 

at 


-r  =  o)I  sin  (tot  +  90) 


0 

(30 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


This  equation  shows  that  the  rate  of  change  -=-  of  an  harmonic 

current  may  be  represented  by  the  projection  *  of  the  line  o>J, 

Fig.  45,   which  is  90°  ahead 
1   of  the  line  I  which  represents 
the  current. 

The  relation  of  i  and  -,-  is 


wt  fixed  line 

*  —  —  —  *  ------- 

Fig.  45. 


most    clearly    shown    by   the 

.  ,.  T-I 

sine  curve  diagram.  Thus 
the  full-line  curve,  Fig.  46,  represents  the  harmonic  cur- 
rent i.  The  steepness  of  this  curve  at  each  point  represents 

the  value  of  -7-.     The  steepness  of  this  curve  is  greatest  at  the 


Fig.  46. 

point  a  where  the  curve  crosses  the  axis,  hence  the  value  of 

-r  is  a  maximum  90°  before  i  reaches  its  maximum.     The  ordi- 

,. 

nates  of  the  dotted  curve  represent  the  values  of  -7-. 

at 

Remark. — It  is  to  be  noted  that  the 
portion  L  -7-  of  the  total  electromo- 
tive force  acting  on  the  circuit,  which 
is  used  to  cause  the  current  to  increase 
Fig.  47.  and  decrease,  is  represented  by  the 

line  coL I,  Fig.  47  ;  the  line  I  represents  the  current  in  the  circuit. 

*  On  a  vertical  fixed  line  not  shown  in  the  figure. 


HARMONIC    ELECTROMOTIVE   FORCE   AND    CURRENT.    57 


49.  Average  or  mean  value  of  an  harmonic  electromotive  force 
or  current.  —  Consider  any  varying  quantity  y.     Its  average  value 


during  an  interval  of  time  from  f  to  t"  is  -,,  —  -,  the  summation 

t    —  t 

being  extended  throughout  the  interval.     That  is, 


Av.y  =  p-^  \ydt 

If  the  successive  values  of  y  be  represented  by  the  ordinates 
of  the  curve,  Fig.  48,  and  the  corresponding  values  of  the  time  t 
be  represented  by  the  abscissas,  then  §  ydt  is  the  area  of  the 

shaded  portion  and  -^ -f  §t,ydt  is  the  height  of  a  rectangle 

t't"dc  of  the  same  area  as  the  curve  and  having  the  same  base. 
The  average  ordinate  of  such  a  curve 
as  Fig.  48  may  be  obtained  quite 
.  closely  by  measuring  the  lengths  of 
a  number  of  equidistant  ordinates. 
The  sum  of  these  ordinates  divided 
by  the  number  of  ordinates  gives  the 
average  ordinate  of  the  curve. 


Proposition. — The  average  value  of  the  sum 
of  a  number  of  quantities  is  equal  to  the  sum  of 
the  average  values  of  each. 


t" 


Fig.  48. 
Proof.  —  Let  x,  y,  z,  ...  be  the  quantities.     Then,  by  definition  we  have 


but 


/"  — 

Therefore 


Av.  (. 


=  A  z>.  x  -J-  Av.  y  -\-   .  . 

z  ...)  =  Av.  x  -f-  Av.  y  -f-  Av.  z-f- 


(33) 
Q.  E.  D. 

50.  Proposition. — The  average  value  of  an  harmonic  electro- 
motive force  or  current  during  half  a  cycle 


is 


a>t  =  o  to  ft)/  =  TT,  or  /  =  o  to  t  =  — 
2  maximum  value 


58  ELEMENTS   OF   ALTERNATING   CURRENTS. 

Proof.  —  Let  <?=  1£  sin  w/  be  the  harmonic  electromotive  force.  Substitute  JE£  sin 
ut  for  y  in  equation  (32)  and  we  have 

E        ft" 

Av.  e  =  -77-  -  :   I     sin  w/  dt 
t"  —  rj? 

Substituting  x  for  w/,  and  remembering  that  the  limits  are  from  w/  =  o  to  ut  =  n,  we 
have 

J5    /»JT  .  X£  f*  21£ 

Av.  e=—    I     smxJx  —  —\     cos  .*  •  =  —  -  (34) 

7T   Jo  ^    |_0  77 

or  the  average  value  of  the  harmonic  electromotive  force  is  twice  the  maximum  value 
divided  by  TT.  Since  —  =  .636,  it  may  also  be  stated  that  the  average  value  of  an 
harmonic  electromotive  force,  or  current,  is  0.636  times  the  maximum  value. 

Remark.  —  The  average  value  of  an  harmonic  electromotive 
force  or  current  during  one  or  more  whole  cycles  is  zero. 

51,  Proposition.  —  The  square  root  of  mean  square,  or  effective 
value,  of  an  alternating  electromotive  force  or  current  during  one  or 

maximum  value 
more  whole  cycles  is  equal  to  —        —  =  —    —  or  o.  707  x  maximum. 

V  2 

Proof.  —  Let  e  =  ]$  sin  w/  be  a  harmonic  electromotive  force.  To  find  the  average 
value  of  <?2  =  E*  sin2  ut  it  is  necessary  to  find  the  average  value  of  the  square  of  the 
sine  of  the  uniformly  variable  angle  ut.  We  have  the  general  relation 

sin2  ut  -\-  cos2  ut  =  I  (a] 

so  that  by  equation  (33) 

Av.  sin2  w/-j-  Av.  cos2  ut=i  (b) 

Now,  the  cosine  of  a  uniformly  variable  angle  passes  similarly  through  the  same  set 
of  values  during  a  cycle  as  the  sine,  hence  Av.  sin2  ut  and  Av.  cos2  ut  are  equal,  so 
that  from  (b]  we  have  : 

2  Av.  sin2  uf  =  I 
or 

Av.  sin2  ut  =  }4 
The  average  value  of  e*  is 

Av.  ^2  =  J$2  Av.  sin2  of 
or 

J?2 

Av.  ^2  =  SL 

2 

and 


=  (35) 

Q.  E.  D. 

Note.  —  The  square  root  of  mean  square  value  of  an  harmonic 
electromotive  force  or  current  is  often  spoken  of  as  the  effective 


HARMONIC   ELECTROMOTIVE   FORCE   AND   CURRENT.    59 

value  of  the  electromotive  force  or  current.  When  it  is  stated 
that  an  alternating  current  is  so  many  amperes,  the  effective  or 
square  root  of  mean  square  value  is  always  meant.  The  same  is 
also  true  with  regard  to  alternating  electromotive  forces.  Here- 
after the  symbol  E  will  be  used  to  designate  the  effective  value 
of  an  electromotive  force  and  /  to  designate  the  effective  value 
of  an  alternating  current.  In  case  the  currents  and  electromotive 
forces  are  harmonic  we  have  the  relations 

j? 

(36) 

/-;%  (37) 

in  which  E  and  I  are  the  maximum  values  of  the  electromotive 
force  and  current  respectively. 

Note. — The  form  factor  of  an  harmonic  electromotive  force 
(see  Article  27)  is  equal  to  ^JJ  or  to  i.i  i. 

52.  Power. — As  pointed  out  in  Article  26,  Chapter  II.,  the 
power  developed  by  an  alternating  electromotive  force  pulsates 
and  in  most  practical  problems  it  is  the  average  power  developed 
which  is  the  important  consideration.  Let  e  =  E  sin  cot  be 
an  harmonic  electromotive  force  acting  on  a 
circuit  and  i  =  I  sin  (cot  —  0)  the  current  pro- 
duced in  the  circuit ;  6  being  the  difference 
in  phase  of  the  electromotive  force  and  current 
as  shown  in  Fig.  49.  The  power  developed 
at  a  given  instant  is  ei  and  in  order  to  estimate  the  average  power 
developed  we  must  find  an  expression  for  the  average  value  of  ei. 

We  have 

ei  =  E I  sin  cot  sin  (cot  —  0) 

or  since  sin  ((ot  —  6)  ==  sin  cot  •  cos  6  —  cos  cot  •  sin  0,  we  have 

ei  =  EI  cos  6  •  sin2  cot  —  EI  sin  6  •  sin  cot  •  cos  cot 
Hence  by  equation  (33) 
Average  ei  =  EI  cos  0  Av.  sin2  (ot  —  EI  sin  0  Av.  sin  cot  cos  cot. 


60  ELEMENTS   OF   ALTERNATING   CURRENTS. 

The  average  value  of  sin  wt  cos  cot  is  zero  since  it  passes  through 
positive  and  negative  values  alike.  The  average  value  of  sin2  col 
is  2.  Therefore 


Average  ei=  Power  =  -  -  cos  0  (38) 

It  is  more  convenient  to  have  this  product  expressed  in  terms 
of  the  effective  values  of  the  current  and  electromotive  force. 
Hence  substitute  for  J^  and  I  their  values  given  by  equations 
(36)  and  (37)  and  we  have 

Power  =  El  cos  0  (39) 

Power  factor.  —  The  factor  cos  6  which  depends  upon  the  in^ 
ductance  and  resistance  of  the  circuit  which  is  receiving  the 
power  (see  Article  54)  is  called  the  power  factor  of  the  circuit. 

Remark.  —  The  expression  for  the  instantaneous  power,  namely 
ei,  may  be  reduced  to  the  form 

/  =  P  —  A  cos  (2wt  —  0) 

in  which  p  is  the  instantaneous  power  ei,  P  is  the  average  power, 
and  A  is  equal  to  P  -=-  cos  6.  Therefore  p  consists  of  a  constant 
part  and  a  part  which  alternates  at  a  frequency  twice  as  great  as 
the  frequency  of  e  and  i. 

Steinmetz  method  of  representing  an  harmonic  alternating  elec- 
tromotive force  or  current.  In  Fig.  16  the  successive  instanta- 
neous values  of  electromotive  force  (and  of  current)  are  repre- 
sented by  the  varying  length  of  a  radius  vector  as  explained  in 
Article  25.  The  curve  traced  by  the  end  of  the  radius  vector  is 
a  circle  when  the  electromotive  force  which  is  represented  is  har- 
monic. Steinmetz  uses  the  diameter  of  this  circle  to  represent  the 
alternating  electromotive  force  ;  and  when,  in  Steinmetz'  s  notation, 
a  current  is  behind  an  electromotive  force  in  phase  the  line  which 
represents  the  current  is  reached  by  a  counter-clockwise  rotation  of 
the  line  which  represents  the  electromotive  force. 

The  proof  of  the  propositions  concerning  the  composition  and 
resolution  of  harmonic  electromotive  forces  and  harmonic  cur- 
rents (see  Article  46)  are  somewhat  more  obscure  in  Steinmetz's 


HARMONIC    ELECTROMOTIVE   FORCE   AND    CURRENT.    6 1 

method  of  representation  than  they  are  in  the  method  of  repre- 
sentation used  in  this  text. 


PROBLEMS. 

32.  An  harmonic  alternating  current  of  65  amperes,  frequency 
133   cycles   per  second,  flows  through  a  circuit.     What  is  the 
maximum  rate  of  change  of  the  current,  and  what  is  the  value  of 
the  current  at  the  instant  of  maximum  rate  of  change  ?     The  cir- 
cuit contains  an  inductance  of  0.08  henry.     What  is   the  maxi- 
mum value  of  the  electromotive  force  which  is  required  to  make 
the  current  change  ?    Ans.  76,800  amperes  per  second,  o  amperes, 
6,144  volts. 

33.  What  is  the  average  value  during  half  a  cycle  of  an  alter- 
nating current  of  maximum  value  10  amperes  ?     This  alternating 
current,  having  a  frequency  of  60  cycles  per  second,  flows  into 
and  out  of  a  condenser.     When  the  current  passes  through  the 
value  zero  the  charge   on  the   condenser  is  +  q,  and  when   the 
current  next  reaches  zero  the  charge  on  the  condenser  is  —  q. 
What  is  the  value  of  q  in  coulombs  ?     What  would  be  the  value 
of  q  if  the  frequency  were  twice  as  great?     Ans.  6.37  amperes, 
0.0264  coulomb,  0.0132  coulomb. 

34.  Two  alternators  A  and  B  are  connected   in  series.     The 
electromotive  force  of  A  is   1,100  volts,  and  the   electromotive 
force  of  B  is  1,200  volts.     The  electromotive  force  of  A  is  90° 
ahead  of  the  electromotive  force  of  B  in  phase.     What  is  their 
combined  electromotive  force  ?     The  two  alternators  give  a  cur- 
rent of  125  amperes  which  lags  30°  behind  their  resultant  electro- 
motive force  in  phase.     What  is  the  power  output  of  each  alter- 
nator?    Ans.  1,628  volts,  29,750  watts,  146,470  watts. 

35.  The   electromotive  force  of  alternator  A,  problem   34,  is 
135  °  ahead  of  the  electromotive  force  of  alternator  B  in  phase. 
A  current  of  1 20  amperes  flows  through  both  alternators,  lagging 
30  °  behind  their  resultant  electromotive  forces.     What  is  the  out- 


62  ELEMENTS   OF   ALTERNATING   CURRENTS. 

put  of  each  alternator?     Ans.   28,750  watts  negative,   121,500 
watts  positive. 

36.  A  harmonic  electromotive  force  e  —  E  sin  wt  produces  in 
a  circuit  a  current  i  —  I  sin  (CD/  —  0) ;  that  is,  this  current  is  6  ° 
behind  the  electromotive  force  in  phase.     Show  that  the  instan- 
taneous power  p  is  equal  to 

p 

P — - b  -  COS  (2ft)/  —  6) 

cos  6 
where  P  is  the  average  power. 

37.  An  alternator  supplies  20  amperes  of  current  at  1, 100  volts 
to  a  receiving  circuit  of  which  the  power  factor  is  0.85  at  a  given 
frequency.     Find  the  values  of  a,  b,  and  0  in  the  expression  : 
instantaneous  power =  a  —  b  cos  (2o>/  —  6).     Ans.    18,700  =  a, 
22,000  =  £,  31.8°  =  6. 

38.  An  alternator  having  a  harmonic  electromotive  force  of 
140  volts  delivers  200  amperes  to  a  circuit  of  which  the  power 
factor  is  0.70.     Find  the  maximum  positive  and  maximum  nega- 
tive values  of  ei.     Ans.  47,600  watts,  —  8,400  watts. 

39.  An  alternator  delivers   200  amperes  of  current  to  glow 
lamps,  and  75  amperes  to  start  an  induction  motor.     The  power 
factor  of  the  motor  while  starting  is  0.3.      Find  the  total  current. 
Ans.   233.7  amperes. 

40.  Two  alternators  each  give  1 20  volts  effective  electromotive 
force  90  °  apart  in  phase.     The  two  machines  are  connected  in 
series  and  they  deliver  200  amperes  of  current  to  a  receiving  cir- 
cuit.    This  current  lags  in  phase  1 5  °  behind  the  resultant  elec- 
tromotive force  of  the  two  machines.     Find  the  total  power  de- 
livered by  the  two  machines,  and  find  the  power  delivered  by 
each.     Ans.  32,775,  12,000,  20,775  watts. 


CHAPTER   V. 
FUNDAMENTAL  PROBLEMS  IN  ALTERNATING  CURRENTS. 

53.  Problem  III.* — To  determine  the  electromotive  force  re- 
quired to  maintain  a  harmonic  alternating  current  in  a  non-induc- 
tive circuit.      Let 

z  =  I  sin  o)t  (a) 

be  the  given  harmonic  current.  The  required  electromotive  force  e 
is  used  wholly  to  overcome  the  resistance  R  of  the  circuit  and  it 
is,  therefore,  equal  to  Ri  so  that 

e  =  R I  sin  <*>t  (b) 

This  electromotive  force  is  harmonic,  its 
maximum  value  is  RI  and  it  is  in  phase  with 
the  current  i.  Thus  the  line  J,  Fig.  50,  rep- 
resents the  given  harmonic  alternating  current 
and  the  line  RI  represents  the  electromotive 
force  required  to  maintain  the  given  current 
Fig  50  in  a  non-inductive  circuit. 

54.  Problem  IV. — To   determine  the   electromotive  force  re- 
quired to  maintain  a  harmonic  alternating  current  in  a  circuit  of 
resistance  R  and  inductance  L.     Let 

i  =  I  sin  tot 

be  the  given  harmonic  current.  The  required  electromotive  force 
consists  of  two  parts,  namely : 

i .  The  part  used  to  overcome  the  resistance  of  the  circuit. 

This  part  is  at  each  instant  equal  to  Ri\  it  is  in  phase  with  i 
and  its  maximum  value  is  RI. 

*  Problems  I.  and  II.  are  given  in  Chapter  I. 

63 


64  ELEMENTS   OF   ALTERNATING   CURRENTS. 

2.  The  part  used  to  make  the  current  increase  and  decrease, 
or  briefly  to  overcome  the  inductance.  This  part  is  at  each  in- 
stant equal  to  L  -r  according  to  equation  (3) ;  it  is  90°  ahead  of 

/  in  phase  (see  Article  48)  and  its  maximum  value  is  coLI.     Let 

the  given  harmonic  alternating  cur- 
rent i  be  represented  by  the  line  J, 
Fig.  5 1 .  Then  RI  is  the  line  which 
represents  Ri\  wLI  is  the  line  which 

di 
represents  L  -y ,  and  the  line  1£  rep- 

resents     the     total     electromotive 
force  required  to  maintain  the  given 
current.      From  the  diagram  we  have 


»2Z2  (40) 

in  which  J?  is  the  maximum  value  of  the  required  electromotive 

force,  and  further 

coL 
tan  °  =  ~R 

in  which   6  is  the  phase  difference   between  the  electromotive 

force  and  current. 

7^ 

The  effective  value  of  the  electromotive  force  is  E  =  — =,  and 

V  2 

the  effective  value  of  the  current  is  /=  -       [by  equations  (36) 

v  2 

and  (37)],  therefore  substituting  */ 2E  for  J?  and  \/2/  for  I  in 
equation  (40),  we  have 

£=/x/^2  + o>2£2  (42) 

When  coL  is  very  small  compared  with  R,  the  effect  of  induc- 
tance is  negligible,  and  this  problem  reduces  to  problem  III. 
When  coL  is  very  large  compared  with  R,  the  angle  0  ap- 
proaches 90°  and  the  power  El  cos  6  becomes  very  small,  al- 
though E  and  /  may  both  be  considerable.  In  this  case  the  cur- 
rent, lagging  as  it  does  90°  behind  the  electromotive  force,  is 


FUNDAMENTAL   PROBLEMS.  65 

called  a  ^wattless  current.     Thus  the  alternating  current  in  a  coil  of 

wire  wound  on  a  laminated  iron  core  is  approximately  wattless. 

Corollary.  —  The  current  which  is  maintained  in  an  inductive 

circuit  by  a  given  harmonic  alternating  electromotive  force  is  a 

E 

current  of  which  the  effective  value  is       —  ^  —  =jj^  by  equation 

(42),  and  which  lags  behind  the  electromotive  force,  by  the  angle 
of  which  the  tangent  is  -~  by  equation  (41). 

Remark.  —  The  relation  between  maximum  values'  of  electro- 
motive force  and  current  (harmonic)  is  in  every  case  the  same  as 
the  relation  between  effective  values,  and  henceforth  effective 
values  will,  as  a  rule,  be  used  in  equations  and  diagrams.  Maxi- 
mum values  will  be  indicated  in  the  text  by  bold-faced  letters, 
*E,  I,  Q,  etc.  ;  effective  values  by  the  letters  E,  /,  etc.,  and  in- 
stantaneous values  by  e,  i,  q,  etc. 

55.  Problem  V.  —  To  determine  the  current  in  an  inductive  circuit  immediately 
after  an  harmonic  electromotive  force,  J£  sin  w/,  is  connected  to  the  circuit. 
The  current  which  can  be  maintained  by  the  given  electromotive  force  is 

sin  (ut—0)  (a) 

* 


according  to  problem  IV.  ;  and  the  decaying  current 


can  exist  in  the  circuit  independently  of  all  outside  electromotive  force,  C  being  a  con- 
stant, as  shown  by  Problem  I.,  Chapter  I.  Therefore  the  current  which  can  exist  in 
an  inductive  circuit  upon  which  an  harmonic  electromotive  force  acts  t  =  if  -f-  i"  or 


"  '     (43) 

in  which  e  is  the  Napierian  base,  ft  is  the  angle  defined  by  equation  (41)  and  C  is  a 
constant.  This  constant  C  is  determined  by  the  condition  that  *  is  equal  to  zero  at 
the  instant  when  the  electromotive  force  is  connected  to  the  circuit.  Let  /'  be  the 
given  instant  at  which  the  harmonic  electromotive  force  begins  to  act  upon  the  circuit 

f  /  —  /'  ) 
Substitute  the  pair  of  values    j  .  j-  in  equation  (43)  and  solve  for  C,  the,  only 

unknown  quantity  ;  then  substituting  this  value  of  C  in  equation  (43)  we  have  the 
expression  for  the  actual  current  which  flows  in  the  circuit  during  the  time  that  the 
maintained  current  is  being  established.  In  a  very  short  time  after  the  electromotive 
force  is  connected  to  the  circuit  the  second  term  of  equation  (43)  disappears  and  the 


66 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


value  of  the  current  at  each  instant  is  given  by  the  first  term,  which  expresses  the  cur- 
rent which  the  given  electromotive  force  can  maintain. 

56.  Problem  VI. — A  coil  of  resistance  R  and  inductance  L,  and 
a  condenser  of  capacity  C  are  connected  in  series  across  alternat- 
ing current  mains  as  shown  in  Fig.  52. 
An  alternating  current  flows  back  and 
forth  through  the  coil  and  charges  the 
condenser  in  one  direction  and  the  other 


main 


Fig.  52.  Fig.  53. 

alternately.  The  problem  of  finding  the  relation  between  the  cur- 
rent in  the  coil  and  the  electromotive  force  between  the  mains  is 
reduced  to  its  simplest  form  as  follows  : 

To  determine  the  electromotive  force  necessary  to  make  the  charge 
q  on  the  condenser  vary  so  that 

q  =  Q  sin  w>t  (a) 

in  which  /  is  elapsed  time,  o>/  is  an  angle  increasing  at  a  constant 
rate,  and  Q  'ls  tne  maximum  value  of  the  charge  in  the  conden- 
ser. This  varying  charge  may  be  represented  by  the  projection 
of  the  rotating  line  Q,  Fig.  53.  The  current  in  the  circuit  is  the 

rate,  ^-9  at  which  the  charge  on  the  condenser  changes.     That 

is  dq 

^  =  }.  (V) 

at 

or  irom  equation  (a)  we  have 

i  =  coQ  cos  wt  (c) 

That  is,  the  current  is  90°  ahead  of  q  in  phase,  its  maximum 
value  is  (oQ  and  it  is  represented  by  the  line  coQ,  Fig.  53.  Using 
the  symbol  I  for  the  maximum  value  of  the  current  we  have 

J=o>Q 


FUNDAMENTAL   PROBLEMS. 


67 


The  required  electromotive  force  is  at  each  instant  used  in  part 
to  overcome  the  resistance  R  of  the  coil,  in  part  to  cause  the 
current  to  increase  and  decrease  in  the  coil,  and  in  part  to  hold 
the  charge  on  the  condenser. 

1 .  The  first  part  is  equal  to  Ri  at  each  instant.      It  is  in  phase 
with  /",  and  its  maximum  value  is  RI. 

2.  The  second  part  is  equal  to  L  --j-  at  each  instant.     It  is  90° 
ahead  of  the  current  and  its  maximum  value  is  toLF. 

3.  The  third  part  is  equal  to  ~  *  at  each  instant.  It  is  in  phase 
with  q  or  90°  behind  the  current,  and  its  maximum  value  is  %? 
or  — -j~t  from  equation \d). 

Let  the  line  J,  Fig.  54,  represent  the  current.  Then  RI  rep- 
resents the  portion  of  the  electromotive  force  used  to  overcome 
resistance,  the  line  &LI  90°  ahead 
of  the  current  represents  the  electro- 
motive force  required  to  overcome 

inductance  and  the  line  —7,  oo  °  be- 

Q)C    ' 

hind  the  current  represents  the  elec- 
tromotive force  required  to  hold  the 
charge  on  the  condenser.  The  line 
H  which  is  the  vector  sum  of  RI, 

o>Z/and  —~  represents  the  total  re- 

0)C 

quired  electromotive  force.     From  the  right  triangle  of  which  the 
sides  are  RI,  wLI  —  —~,  and  U,  we  have 


0 


RI 


Fig.  54. 


or 


(44) 


*  Since  g=Cf  according  to  equation  (15),  Chapter  I. 


68  ELEMENTS   OF   ALTERNATING   CURRENTS. 

and  T        1 

coL p,  ,     N 

c*C  (45) 


tan  6 


R 


Corollary. — A  given  harmonic  electromotive  force  acting  upon  a 
circuit  containing  a  condenser  of  capacity  C,  inductance  L  and 
resistance  R  maintains  a  current  of  which  the  effective  value  -is 


(46) 


and  which  lags  behind  the  electromotive  force  by  the  angle  0 
defined  by  equation  (45).    The  quantity  a>L—  —^  may.  be  either 

positive  or  negative  according  as  coL  or  —  _  is  the  greater  so  that 

the  current  may  be  either  behind  or  ahead  of  the  electromotive 
force  in  phase.     In  fact  the  limiting  values  of  6  are  db  90°. 

Connection  of  condensers  in  series.  —  When  a  current  /  flows 
through  two  condensers  in  series  the  electromotive  forces  IjwC 
and  f/(oCf  are  in  phase  with  each  other,  so  that  the  total  electro- 
motive force  required  to  overcome  the  reaction  of  the  two  con- 
densers is 


The  oscillatory  current.  —  If  (oL  --  -~  =  o  then  the  impressed 

electromotive  force  has  only  to  overcome  the  resistance  of  the  cir- 
cuit, and  problem  VI.  reduces  in  form  to  problem  III.  If  the 
resistance  of  the  circuit  in  this  case  were  negligibly  small  then  no 
electromotive  force  at  all  would  be  required  to  maintain  the  given 
harmonic  current.  Such  a  self-sustained  harmonic  current  is 

called  an  oscillatory  current.  In  this  case,  from  coL  --  -~  =  o,  we 
have 


FUNDAMENTAL    PROBLEMS.  69 

or  since  a)  —  2irf  [equation  (28)]  we  have 


This  equation  expresses  what  is  called  the  proper  frequency  of 
oscillation  of  the  inductive  circuit  of  a  condenser.  In  case  the 
resistance  of  the  circuit  is  not  zero,  which  is  of  course  the  only 
real  case,  then  the  only  current  which  can  exist  in  the  circuit  in- 
dependently of  any  impressed  electromotive  force  is  a  decaying 
oscillatory  current  the  discussion  of  which  is  beyond  the  scope  of 
this  text.  The  character  of  this  decaying  oscillatory  current  is 
shown  by  the  curve,  Fig.  55.  The  ordinates  of  this  curve  rep- 
resent the  successive  values  of  the  current  produced  when  a 
charged  condenser  is  discharged  through  an  inductive  circuit. 


\ 


Fig.  55. 

57.  Electric  resonance. — By  inspecting  equation  (46)  we  see 
that  an  electromotive  force  of  given  effective  value  E  produces 
the  greatest  current  in  the  circuit,  Fig.  52  (R,  L  and  C  given), 

when  the  frequency  is  such  that  coL ~  is  zero.  This  produc- 
tion of  a  greatest  current  by  a  given  electromotive  force  at  a  crit- 
ical frequency  is  called  electric  resonance.  Thus  the  ordinates  of 
the  curve,  Fig.  56,  represent  the  values  of  effective  current  at 
various  frequencies  (abscissas).  The  curve  is  based  on  the 
values  E=  200  volts,  R  =  2  ohms,  L  =  .352  henry  and  C=  20 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


microfarads.  The  maximum  point  of  the  curve  is  not  a  cusp,  as 
would  appear  from  the  figure,  but  the  maximum  is  so  sharply 
defined  that  it  cannot  be  properly  represented  in  so  small  a  fig- 
ure. When  the  frequency  of  the  electromotive  force  is  zero, 


100 


100         110 


w  10  £0         30         40          50         60         70 

Frequency 

Fig.  56. 

which  is  the  case  when  a  continuous  electromotive  force  acts  on 
the  circuit,  the  current  is  zero  except  for  the  very  slight  current, 
which  is  conducted  through  the  dielectric  between  the  condenser 
plates.  When  the  frequency  of  the  electromotive  force  is  very 
great  the  current  approaches  zero,  inasmuch  as  a  very  small  cur- 
rent of  high  frequency  must  increase  and  decrease  at  a  very  rapid 
rate,  and  to  produce  this  rapid  increase  and  decrease  a  very  great 
electromotive  force  is  required.  At  low  frequency  the  current  is 
kept  down  in  value  by  the  condenser,  and  at  high  frequency  the 
current  is  kept  down  in  value  by  the  inductance. 

Remark  i. — At  critical  frequency,  &L -~  =  o  and  equation 

(46)  becomes  simply  f  =  ~. 

Remark  2. — While  an  electromotive  force  is  being  established 
between  the  plates  of  a  condenser  the  dielectric  is  subjected  to  an 
increasing  electrical  stress  and  this  increasing  electrical  stress  is 
exactly  equivalent,  in  its  magnetic  action,  to  an  electric  current 
flowing  through  the  dielectric  from  plate  to  plate.  Magnetically, 
therefore,  a  circuit  containing  a  condenser  is  a  complete  circuit. 


FUNDAMENTAL   PROBLEMS.  71 

Increasing  (or  decreasing)  electrical  stress  is  called  displacement 
current. 

Multiplication  of  electromotive  force  by  resonance. — When  reso- 
nance exists  in  a  circuit  containing  an  inductance  and  a  condenser 
in  series,  the  alternating  electromotive  force  toLI  between  the 
terminals  of  the  inductance,  and  the  alternating  electromotive 
force  f/coC  between  the  terminals  of  the  condenser,  may  each  be 
much  greater  than  the  alternating  electromotive  force  which  acts 
upon  the  circuit.  This  fact  is  easily  understood  by  means  of  the 
mechanical  analogue.  If  even  a  very  weak  periodic  force  act 
upon  a  weight  which  is  suspended  by  a  spiral  spring,  the  weight 
will  be  set  into  violent  vibration,  provided  the  frequency  of  the 
force  is  the  same  as  the  proper  frequency  of  oscillation  of  the 
body.  The  forces  acting  on  the  spring  may  reach  enormously 
greater  values  than  the  periodic  force  which  maintains  the  motion 
of  the  system.  Also  \h&  forces  which  act  upon  the  weight  to  pro- 
duce its  up  and  down  acceleration  may  greatly  exceed  in  value 
the  periodic  force  which  maintains  the  motion. 

Example. — A  coil  of  0.352  henry  inductance  and  2  ohms  re- 
sistance and  a  condenser  of  20  microfarads  capacity,  are  connected 
in  series  between  alternating  current  mains.  The  critical  fre- 
quency of  this  circuit  is  60  cycles  per  second,  according  to  equa- 
tion (48).  The  electromotive  force  between  the  mains  is  200 
volts  and  its  frequency  is  60  cycles  per  second.  The  current  in 
the  circuit  is  100  amperes,  according  to  equation  (46),  and  the 
effective  electromotive  force  between  the  condenser  terminals  is 
13,270  volts  (=  f/o)C). 

Multiplication  of  current  by  resonance. — When  resonance  exists 
in  a  circuit  containing  an  inductance  and  a  condenser  in  parallel, 
as  shown  in  Fig.  57,  the  alternating  current  in  each  branch  may 
greatly  exceed  in  value  the  alternating  current  /,  which  is  deliv- 
ered to  the  system.  Let  OE,  Fig.  58,  represent  the  electromo- 
tive force  between  the  branch  points  a  and  b,  Fig.  57  ;  that  is, 
the  electromotive  force  of  the  alternator.  The  current  I.2  in  the 
condenser  is  nearly  90°  ahead  of  E  in  phase,  and  the  current  /p 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


in  the  inductance,  is  nearly  90°  behind  E  in  phase,  so  that  the 
resultant  of  1^  and  I2  is  small,  as  shown  in  Fig.  58. 


Fig.  57. 


Fig.  58. 


This  multiplication  of  current  by  resonance  is  easily  under- 
stood by  means  of  the  mechanical  analogue  as  follows  :  A  lever 
suspended  at  the  center  carries  a  weight  L  on  one  end,  while  the 


Fig.  59. 


other  end  is  held  down  by  a  spiral  spring  C.  If  the  center  of 
the  lever  is  moved  up  and  down  to  a  very  slight  extent  at  the 
proper  frequency,  the  system  will  be  set  into  violent  vibration 


FUNDAMENTAL   PROBLEMS.  73 

and  the  velocities  II  and  72  of  the  ends  of  the  lever  will  greatly 
exceed  in  value  the  velocity  /  of  the  center  of  the  lever,  although 
half  the  sum  of  the  velocities  of  the  ends  of  the  lever  is  at  each 
instant  equal  to  the  velocity  of  the  center  of  the  lever,  j  ust  as  the 
sum  of  the  currents  7L  and  /2,  in  Fig.  58,  is  at  each  instant  equal 
to  the  current  /. 


PROBLEMS. 

41.  A  circuit  has  inductance  L=o.2  henry  and  a  resistance 
R  =  6  ohms.      Calculate  the  current  produced  by  100  volts,  the 
frequency  being  60  cycles  per  second.     Calculate  the  phase  dif- 
ference between  the  electromotive  force  and  current.     Calculate 
power  developed.     Ans.  1.325  amperes,  85°  27',  10.84  watts. 

42.  A  circuit  has  160  ohms  resistance  and  0.2  henry  inductance. 
Calculate  the  power  factor  of  the  circuit  for  a  frequency  of  60 
cycles  per  second.     Ans.  0.9046. 

43.  An  electromotive  force  of  20,000  volts  acts  on  a  receiving 
circuit  of  which  the  power  factor  is  0.85.     Find  the  component 
of  electromotive  force  parallel  to  the  current  and  the  component 
of  electromotive  force  perpendicular  to  the  current.     Ans.  17,000 
volts,  10,536  volts. 

44.  Show  that  El  cos  6  =  RP  in  a  circuit  of  resistance  R,  of 
inductance  Z,  and  containing  a  condenser  of  capacity  C. 

Suggestion. — Substitute  in  El  cos  6  the  value  of  E  in  terms  of 
7,  R,  L,  C,  and  &>,  and  the  value  of  cos  6  in  terms  of  R,  L,  C, 
and  ft>. 

45.  A  non-inductive  resistance  of   20  ohms,  a  resistanceless 
inductance  of  0.06  henry,  and  a  condenser  of  105   microfarads 
capacity  are  connected    in   series   to    no-volt  6o-cycle    mains. 
Find  the  electromotive  force  between  the  terminals  of  the  resist- 
ance, between  the  terminals  of  the  inductance,  and  between  the 
terminals  of  the  condenser.      Find  the  same  electromotive  forces 
when  the  circuit  is  connected  to  I  lo-volt  i2O-cycle  mains.     Ans. 


74  ELEMENTS   OF   ALTERNATING   CURRENTS. 

(a)   109.06  volts,    123.3   volts,  138  volts;   (fr)   57.5   volts,  130.1 
volts,  126.3  volts. 

46.  An  inductance  of  o.  I  henry  and  a  resistance  R  are  con- 
nected in  series  to  no-volt  6o-cycle  mains.      Calculate  the  cur- 
rent values  when  R  equals  zero,  when  R  equals  one  ohm,  when 
R  equals   5  ohms,  when  R  equals   10  ohms,  when  R  equals  20 
ohms,  when  R  equals   50  ohms,  and  when  R  equals   100  ohms, 
and  plot  a  curve  of  which  the  abscissas  represent  R  and  the  ordi- 
nates  represent  corresponding  values  of  the  current.     Ans.  2.919 
amperes,  2.917   amperes,   2.893   amperes,   2.821   amperes,  2.578 
amperes,  1.757  amperes,  1.029  amperes. 

47.  A  circuit  contains  a  constant  inductance  and  a  variable  re- 
sistance.    Show  in  general  that  the  current  produced  in  the  cir- 
cuit by  a  given  electromotive  force  is  nearly  independent  of  the 
resistance  so  long  as  the  resistance  is  small  compared  with  coL. 

48.  A  circuit  contains  a  constant  resistance  and  a  variable  in- 
ductance.    Show  in  general  that   the   current  produced  in  the 
circuit  by  a  given  electromotive  force  is  nearly  independent  of  the 
inductance  so  long  as  coL  is  small  compared  with  R. 

49.  Alternating  current  mains  deliver  100  amperes  of  current 
to  a  non-inductive  circuit,  to  glow  lamps  for  example.     An  in- 
ductive circuit  of  negligible  resistance  is  then  connected  to  the 
mains  and  it  takes  10  amperes.      What  is  the  total  current  de- 
livered by  the  mains?     Ans.  100.5  amperes. 

50.  Two  condensers,  each  of  negligible  resistance,  have  capac- 
ities  of  0.5   and    0.05   microfarad  respectively.     The   two  con- 
densers are  connected  in  series  to  i,ioo-volt  alternating  current 
mains.     What  is  the  electromotive  force  between  the  terminals 
of  each  condenser?     Ans.  100  volts,  1,000  volts. 

5  i.  An  electrostatic  voltmeter  has  a  capacity  of  0.0006  micro- 
farad when  its  deflection  is  a  and  0.0008  microfarad  when  its 
deflection  is  b.  The  electromotive  force  between  the  terminals 
of  the  instrument  is  75  volts  when  its  deflection  is  a,  and  125 
volts  when  its  deflection  is  b.  An  auxiliary  condenser  of  0.007 


FUNDAMENTAL   PROBLEMS.  75 

microfarad  capacity  is  connected  in  series  with  the  instrument. 
Find  electromotive  forces  necessary  to  produce  deflections  a  and 
b.  Ans.  8 1. 4  volts,  139.3  volts. 

52.  A  direct-reading    electrostatic    voltmeter    having  0.0006 
microfarad  capacity  is  connected  through  50,000  ohms  non-in- 
ductive resistance  to  60  cycle  mains.     Find  the  percentage  error  * 
of  the  readings  of  the  instrument  and  state  whether  it  indicates 
too  high  or  too  low.     Ans.  0.0064  per  cent,  too  low. 

53.  The  above  electrostatic  voltmeter  is  connected  in  series 
with  an  inductance  of  2  henrys,  resistance  negligible.      Find  the 
percentage  error  of  the  voltmeter  readings  when  a  frequency  of 
60  cycles  is  used,  and  state  whether  it  indicates  too  high  or  too 
low.     Ans.  0.0017  per  cent,  too  high. 

54.  An  electrometer  having  an  inductance  of  0.02  henry  and 
a  resistance  of  1,500  ohms  gives  the  same  deflection  for  a  certain 
6o-cycle  electromotive  force  as   it  does  for    127.5   volts   direct 
electromotive  force.     What  is  the  effective  value  of  the  6o-cycle 
electromotive  force  ?     Ans.    127.5017  volts. 

55.  An  electrodynamometer   has  a   resistance   of  500  ohms. 
When  used  as  an  alternating  voltmeter  at  a  frequency  of  60 
cycles  per  second  its  percentage  error  is  -fa  per  cent.     What  is 
its  inductance?     Ans.  0.0594  henry. 

*  In  this  and  subsequent  problems  the  percentage  error  of  an  instrument  is  under- 
stood to  mean  the  difference  between  the  instrument  reading  and  the  true  value  of  the 
quantity  measured,  divided  by  the  true  value  of  the  quantity. 


CHAPTER  VI. 

THE   USE   OF   COMPLEX   QUANTITY.     STEINMETZ'S 
METHOD. 

58.  Methods  in  alternating  currents.     The  graphical  method  and 
the  trigonometrical  method, — All  fundamental  problems  *  in  alter- 
nating currents  may  be  solved  by  the  graphical  method,  in  which 
the  various  electromotive  forces,  currents,  etc.,  are  represented  by 
lines  in  a  diagram  and  the  required  results  are  measured  off  as 
in  graphical  statics.     In  practical  problems,  however,  the  different 
quantities  under  consideration  differ  so  greatly  in  magnitude  that 
it  is  difficult  to  scale  off  results  with  any  degree  of  accuracy. 
The  graphical  method  is,  however,  particularly  useful  for  giving 
clear  representations,  and  trigonometric  formulas  may  be  used  in 
connection  with  graphical  diagrams  in  every  case. 

The  trigonometric  formulas  in  the  more  complicated  problems, 
however,  become  very  unwieldy  and  are  not  suitable  for  easily 
obtaining  numerical  results. 

Steinmetz's  method. — Numerical  results  in  alternating  current 
calculations  are  most  easily  obtained  by  means  of  a  method  de- 
veloped mainly  by  Steinmetz,  in  which  complex  quantity  is  used. 
This  method  is  purely  algebraic  and  is  called  by  Steinmetz  the 
symbolic  method. 

59.  Simple  quantity.     Complex  quantity. — A  simple  quantity 
is  a  quantity  which  depends  upon  a  single  numerical  specification. 

*  The  fundamental  problems  are  those  which  treat  of  harmonic  electromotive  force 
and  harmonic  current.  It  is  a  mistake  to  suppose  that  differential  equations  furnish  a 
method  for  treating  alternating  currents  distinct  from  the  three  methods  mentioned 
above.  In  the  application  of  differential  equations  the  first  step  is  to  derive  one  or 
more  harmonic  expressions  for  electromotive  force  and  current  and  the  subsequent 
development  is  precisely  the  one  or  the  other  of  the  above-mentioned  methods. 

76 


THE   USE   OF   COMPLEX   QUANTITY.  77 

Simple  quantities  are  often  called  scalar  quantities.  A  complex 
quantity  requires  two  or  mgre  independent  numerical  specifica- 
tions to  entirely  fix  its  value.  For  example,  if  wealth  depends 
upon  the  possession  of  horses  (h)  and  cattle  (r),  then,  if  no 
agreement  exists  as  to  the  relative  value  of  horses  and  cattle  (in 
fact  any  such  agreement  is  essentially  arbitrary),  the  specification 
of  the  wealth  of  an  individual  would  require  the  specification  of 
both  horses  and  cattle.  Thus  the  wealth  of  an  individual  might 
be  5/5  +  loor.  The  two  or  more  numbers  which  go  to  make  up 
a  complex  quantity  are  called  the  elements  of  the  quantity. 

Addition  and  subtraction.  —  Two  complex  quantities  are  added 
or  subtracted  by  adding  or  subtracting  the  similar  numerical 
elements  of  the  quantities.  Thus  5/2  +  icor  added  to  6h  -f  I5r 
gives  \\h-\-  1  1  5r,  and  2/1  +  2$c  subtracted  from  5//  -f  IOOT  gives 


Multiplication  and  division.  —  Consider  two  complex  quantities 
5/j  -f  2c  and  3/2  -f  4<r  in  which  //  and  c  are  independent  incommen- 
surate units,  say  horses  and  cattle.  Multiplying  the  first  of  these 
expressions  by  'the  latter,  using  the  ordinary  rules  of  algebra  ,  we 

have  : 

5/*  +  2c  (3/2  -f  4c  =  1  5/*2  -f  6/ie  +  20c/i  +  Sr2 


Now  in  general  the  squares  and  products  of  units  c2,  fr,  cli  and 
he  have  no  meaning  ;  so  that  the  significance  of  the  product  of 
two  complex  quantities  depends  upon  arbitrarily  chosen  meanings 
for  these  squares  and  products  of  units. 

60.  Vectors.  —  A  vector  is  a  quantity  which  has  both  magni- 
tude and  direction.  A  vector  may  be  specified  by  giving  its  com- 
ponents in  the  direction  of  arbitrarily  chosen  axes  of  reference. 
In  specifying  a  vector  by  its  components,  it  is  necessary  to  have 
it  distinctly  stated  which  is  its  x  component  and  which  is  its  y 
component.*  This  may  be  done  either  by  verbal  statement  or 
by  marking  one  of  the  components  by  a  distinguishing  index. 
Further  it  is  allowable  to  connect  the  two  components  with  the 

*  \Ve  are  at  present  concerned  only  with  vectors  in  one  plane. 


78  .  ELEMENTS   OF   ALTERNATING   CURRENTS. 

sign  of  addition.     Thus  a  -\-jb  is  a  specification  of  the  vector 
of  which  the  x  component  is  a  and.  the  y  component  is  b  ;  the 

index  letter  j  being  used  to  mark  the 
y  component.     This  expression  of  a 
vector  is  a  complex  quantity,  the  in- 
dependent units  of  which  are  vertical 
and    horizontal   distances  or  compo- 
^    nents.     For  example,  the  vector  OP, 
Fig.  60,  is  specified  by  the  horizontal 
component  a  and  the  vertical  compo- 
Fig-  60>  nent  b.    The  vector  may  therefore  be 

specified  by  the  expression  a  +jb  the  index  y  being  used  to  show 
that  b  is  the  vertical  component. 

Numerical  value  and  direction  of  a  vector. — The  numerical 
value  of  a  vector  is  the  square  root  of  the  sum  of  the  squares  of 
its  components.  Thus  the  numerical  value  of  the  vector  a  -f  jb 


o 


a 


is  N/      +     . 

The  angle  between  the  vector  and  the  x  axis   is  the  angle  0, 

of  which  the  tangent  is  -  >   that  is  tan  6  =  This  matter  of 

a  a 

value  and  direction  of  a  vector  is  an  important  consideration  in 
alternating  current  problems. 

Addition  and  subtraction  of  vectors.  —  The  sum  of  a  number  of 
vectors  is  a  vector  of  which  the  x  component  is  the  sum  of  the 
x  components  of  the  several  vectors,  and  of  which  the  y  com- 
ponent is  the  sum  of  the  y  components  of  the  several  vectors. 
Thus  the  sum  of  the  vectors  a  +  jb,  a'  +jbf  ',  a"  -\-jbff  is 


The  difference  of  the  two  vectors  a  -\-jb,  a'  +jbf  is 


Multiplication  of  vectors.  —  Consider  the  two  vectors  a  +jb  and 
a'  +jb'.  Multiply  these  two  expressions,  using  the  formal  rules 
of  algebra,  and  we  have  for  the  product 

aaf  +jabf 


THE   USE   OF   COMPLEX   QUANTITY.  79 

Each  term  in  this  product  must  be  interpreted  arbitrarily  or 
according  to  convention  in  order  that  this  product  may  have  a 
definite  meaning. 

In  the  first  place,  we  may  take  aaf  ,  which  is  not  affected  by 
the  index  j\  to  be  a  horizontal  quantity,  and  we  may  take  ab' 
and  a'b  to  be  vertical  quantities  or  vertical  components  of  a  vec- 
tor. As  to  the  term  j2bb'  we  may  note  that  the  index  letter  /, 
used  once,  indicates  that  a  quantity  is  vertical,  while  without  the 
index  j  the  quantity  would  be  understood  to  be  horizontal  and 
to  the  right  (see  Fig.  60)  ;  that  is,  the  letter  j  may  be  thought  of 
as  turning  a  quantity  through  90°  in  the  positive  direction,  that 
is,  counter-clockwise.  It  is,  therefore,  convenient  to  think  of 
the  letter  /when  used  twice  (jjbb'  or  j2bb'}  as  turning  the  quan- 
tity upon  which  it  operates  through  180°,  or  as  reversing  its  di- 
rection, and,  therefore,  its  algebraic  sign.  That  is,  j2bb'  is  to  be 
taken  as  equal  to  —  bb'  or 

/--i 

so  that  the  product  of  the  two  vectors  a  +  ja'  and  b  +  jb'  is  to 
be  interpreted  as  the  vector 

(ab  -  a'b'}  +j(ab'  +  a'b  ) 
Quotient  of  two  vectors.  —  Consider  the  quotient 

a  +  jb 

a'  +jb> 

multiply  both  numerator  and  denominator  by  a1  —jb',  remem- 
bering thaty2  =  —  1,  and  we  have 

a  +  jb       aa'  +  bb'        .  (a'b  —  ab' 


.  (a' 
+  J  \a' 


a'  +  jb'       a'2  + 

which  leads  to  the  conception  of  the  quotient  of  the  two  vectors 
as  a  vector  of  which  the  x  component  is 

aa'+  bb' 
a'2  +  b'2 
and  the  y  component  is 

a'b-ab' 
a"  +  b'2 


So  ELEMENTS   OF   ALTERNATING   CURRENTS. 

61.  The  fundamental  equations  in  the  application  of  complex 
quantity  to  alternating  current  problems.  —  The  problem  which 
forms  the  basis  of  nearly  every  discussion  of  alternating  current 
phenomena  is  problem  VI.,  which  is  discussed  in  Article  56. 
This  problem  deals  with  the  relation  between  electromotive  force 
and  current  in  a  circuit  of  resistance  r,  of  inductance  L  and  con- 
taining a  condenser  of  capacity  C.  The  electromotive  force  E 
has  a  component,  rl,  which  is  in  phase  with  the  current  /,  and  a 

component  f  o)L  --  -^  j  /,  which  is  90°  ahead  of  the  current  in 

phase.  Therefore,  using  the  current  vector  as  the  reference 
axis,  we  have  the  following  complex  expression  for  the  electro- 
motive force  : 


(49) 

This  equation  is  so  often  used  that  it  is  convenient  to  represent 
the  factor  j  a>L -^  |  by  the  single  symbol  x.     That  is 


Using  this  abbreviation,  equation  (49)  becomes 


or 

E-(r+jz)I  (5!) 

The  complex  quantity  {r  +jx}  is  called  the  impedance  of  the 
circuit.  The  factor  x  is  called  the  reactance  of  the  circuit. 

Resistance  and  reactance  are  both  expressed  in  ohms.  The 
numerical  value  of  impedance,  namely,  %/r2  -f  ^2,  is  expressed  in 
ohms. 

When  a)L  is  greater  than  l/o>£7  the  reactance  x  is  positive. 
When  1/co  C  is  greater  than  coL  the  reactance  is  negative.  When 
reactance  is  positive  the  component  of  E  which  is  at  right  angles 
to  /  is  90°  ahead  of  /  in  phase.  When  reactance  is  negative  the 
component  of  E  which  is  at  right  angles  to  /  is  90°  behind  /  in 
phase. 


THE   USE   OF   COMPLEX   QUANTITY.  8 1 

In  some  problems  it  is  convenient  to  take  the  electromotive 
force  vector  as  the  reference  axis,  and  to  express  current  in  terms 
of  its  components  parallel  tp  and  perpendicular  to  E  respectively. 
Thus  solving  equation  (51)  for  /we  have 

~ 
or 


or 

The  complex  quantity  j  ~YJ~ — 2 — /•    a  . — \  f  1S  called  the  admittance  of  the  circuit. 

The  quantity  j  g  I  ,  by  which  E  is  multiplied  to  give  the  component  of  7 

parallel  to  Et  is  called  the  conductance  of  the  circuit.  ^ 

The  quantity  j  g  I  ,  by  which  ^  is  multiplied  to  give  the  component  of  / 

perpendicular  to  E  is  called  the  susceptance  of  the  circuit. 
Equation  (52)  is  sometimes  written  : 

I=(g-jb}E  (53) 

in  which  g  is  the  conductance  of  the  circuit  and  b  is  its  susceptance. 

Remark. — Algebraic  developments  based  upon  equations  (5 1 ) 
and  (53)  always  give  electromotive  forces  and  currents  in  terms  of 
their  rectangular  components. 

When  a  numerical  value  is  required  the  square  root  of  the 
sum  of  the  squares  of  the  components  must  be  taken. 

When  the  phase  angle  between  a  calculated  electromotive 
force  or  current  and  the  reference  axis  is  desired  it  is  found  as 
the  angle  whose  tangent  is  equal  to  the  ratio  of  the  components. 

62.  Expression  for  power. — Consider  an  electromotive  force 

which  maintains  a  current 

I=b+jb'  (ii) 

in  a  circuit.     The  power  developed  by  the  electromotive  force  is 


82      ELEMENTS  OF  ALTERNATING  CURRENTS. 

equal  to  the  product  of  the  numerical  values  of  E  and  /  into  the 
cosine  of  the  angular  phase  difference  between  E  and  /  as  ex- 
plained in  Article  52.  When  E  and  /  are  given  in  terms  of  their 
rectangular  components,  as  in  equations  (i)  and  (ii)  above,  it  is 
more  convenient  to  calculate  power  by  means  of  the  formula 

P=ab  +  a'bf  (54) 

that  is,  the  power  is  equal  to  the  product  of  .^-components  of 
electromotive  force  and  current  plus  the  product  of  j-compo- 
nents  of  electromotive  force  and  current.  When  a  and  b  are  op- 
posite in  direction  their  product  is  negative,  and  when  a'  and  br 
are  opposite  in  direction  their  product  is  negative. 

PROBLEMS. 
56.  Separate  the  components  of  the  complex  expression 


find  the  numerical  value  of  the  expression,  and  find  the  tangent 
of  the  angle  between  it  and  the  axis  of  reference,  each  in  terms 
of  a,  b,  c,  d,  e  and/". 

57.  A  current  of  which  the  components,  referred  to  an  arbi- 
trary axis  of  reference,  are  25  amperes  and  —  10  amperes  respec- 
tively, or  of  which  the  complex  expression  is  25  —  iq/,  flows 
through  a  circuit  of  which  the  resistance  is  8  ohms  and  the  re- 
actance is  -f  1  2  ohms.      Find  the  components  of  the  electromo- 
tive force  which  maintains  the  current  and  find  the  power  devel- 
oped.     Calculate  the  power  by  the  use  of  equation  (54)  and  also 
by  multiplying  the  resistance  of  the  circuit  by  the  square  of  the 
numerical  value  of  the  current.     Ans.  x  component  =320  volts, 
y  component  =220  volts,  5,800  watts. 

58.  A  circuit  has  5  ohms  resistance  and  -f-  3  ohms  reactance. 
What  is  the  numerical  value  of  the  impedance  in  ohms  ?     What  is 
the  inductance  of  the  circuit  in  henrys,  the  frequency  being  133 


THE   USE   OF   COMPLEX   QUANTITY.  83 

cycles  per  second,  no  condenser  being  connected  in  the  circuit  ? 
Ans.  6  ohms,  0.00359  henry. 

59.  An   electromotive  force  of  which  the  components  are  50 
volts  and  75  volts  respectively,  that  is,  the  electromotive  force  is 
expressed  by  50  -f-  757,  produces  a  current  through  a  circuit  of 
which  the  resistance  is  10  ohms  and  the  reactance  is  —  6  ohms. 
Find  the  components  of  the   current,   and   find  the   power  de- 
veloped.    Calculate  the  power  by  the  use  of  equation  (54)  and 
also  by  multiplying  the  resistance  of  the  circuit  by  the  square  of 
the  numerical  value  of  the  current.     Ans.  .^-component  of  cur- 
rent =  0.368  amperes,  ^-component  of  current  =  7.72  amperes, 
power  =  597.4  watts. 

60.  Separate  the  components  of  the  complex  expression 

* 

E=  r.L  +  ?(0LJ.  -f 

1    l 


61.  Separate  the  components  of  the  complex  expression  ejy. 
Ans.  ejy  =  cos  y  -\-j  sin  y. 

Suggestion.  —  Look  up  the  series  for  e*.  Substitute  jy  for  x  in 
this  series.  Separate  real  and  imaginary  terms  into  two  series, 
and  compare  these  series  with  the  series  for  cos  y  and  sin  y. 

62.  Show  that  £(cos  6  +j  sin  0)  is  6°  ahead  of  E  in  phase  and 
that  its  numerical  value  is  equal  to  the  numerical  value  of  E. 


CHAPTER   VII. 


FURTHER   PROBLEMS. 
APPLICATION   OF   THE   SYMBOLIC   METHOD 

63.  Problem  VII.  Coils  in  series. — An  alternating  electromo- 
tive force  E  acts  upon  two  coils  connected  in  series  as  shown  in 
Fig.  6 1.  It  is  required  to  find  El  and  E2  each  in  terms  of  Ey  rv 


O( 


Fig.  61. 


Fig.  62. 


r2,  xv  and  x2.  Let  /  be  the  current  in  the  circuit.  The  general 
relation  between  E,  Ev  E2,  and  /  is  shown  in  Fig.  62.  This 
figure  is  given  in  order  that  the  following  equations  (a\  (fr),  and 
(c)  may  be  more  easily  understood.  According  to  equation  (51) 
we  have  : 

E,  -  (r,  +>,)/  (a) 

and 


Further,  the  total  electromotive  force  E  is  the  resultant  of  El  and 
E2  as  shown  in  Fig.  62,  so  that 


APPLICATION   OF   THE   SYMBOLIC   METHOD.  85 

This  is  of  course  a  complex  equation.     Substituting  the  values 
of  .Z^  and  £2  from  (a)  and  (&)  in  (c)  and  solving  for  /,  we  have 


Substituting  this  Value  in  (a)  and  (£)  we  have 

(r.  +  A)^ 


,,,•,) 

and 

E  ___    +      E 

~ 


These  equations  (e)  and  (/)  express  the  electromotive  forces  El 
and  EI  in  terms  of  the  known  quantities  E,  rv  r2,  x^  and  xy  For 
purposes  of  numerical  calculation  (e)  [and  likewise  (/)]  must  be 
separated  into  components,  that  is,  into  real  and  imaginary  parts, 
and  the  numerical  value  of  Ev  (and  likewise  of  E^  is  then  found 
by  taking  the  square  root  of  the  sum  of  the  squares  of  these 
components.  Thus,  multiplying  numerator  and  denominator  of 
(e)  by  r^  -j-  r2  —  j(x^  +  ;r2),  we  remove  j  from  the  denominator, 
and  may  then  separate  the  components,  namely, 


p  _ 


-f 


The  first  term  of  this  expression  is  the  component  of  ^  par- 
allel to  E  and  the  second  term,  dropping  j,  is  the  component  of 
E^  perpendicular  to  E,  and  the  numerical  value  of  El  is  the 
square  root  of  the  sum  of  the  squares  of  these  components.* 
The  final  result  is  of  no  use  whatever  in  giving  a  conception  of 
the  phenomenon  under  consideration.  It  is  useful  only  when 
it  is  desired  to  carry  out  numerical  calculations.  It  is,  indeed, 
generally  the  case  that  the  use  of  the  symbolic  method  in  the 
solution  of  the  alternating  current  problems  is  simple  and  instruc- 
tive in  its  initial  steps  only,  while  the  final  solution  itself  is  unin- 


(numerical  value)  = 


86 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


telligible.  The  final  results  will,  therefore,  be  written  out  in  full 
only  when  the  student  is  expected  to  use  them  in  numerical  cal^ 
culations. 

The  following  simple  cases  of  the  problem  under  consideration 
are  particularly  interesting. 

1.  When  x^  =  o  and  x2  =  o  ;  then  £,  Ev  E2,  and  /  are  parallel 
and  E  =  El  -f  £2  (numerically). 

x       x 

2.  When  —  =  — ;   then  E,  Ev   and  E2  are   parallel   to  each 

r\       ri 
other  and  all  are  ahead  of  /  in  phase,  by  the  angle  of  which  the 

tangent  is  — .     In  this  case,  also,  E  —  El  -f  E2  (numerically). 
r\ 
x  x 

3.  When  —  is  very  small  and  —  very  large;  then  El  is  parallel 

r\  r-z 

to  /  and  E2  is  at  right  angles  to  7,  as  shown  in  Fig.  63.     This 

figure  is,  of  course,  a  particular  case  of  Fig.  62.  In  the  present 
case  the  numerical  relation  between  E,  El  and  E2  is 


and  when  El  is  small  E2  is  sensibly  equal  to  E,  numerically. 

Example. — A  transmission  line  of  large  resistance  r^  and  small 
reactance  x^  supplies  current  to  a  receiving  circuit  of  large  react- 

x  x 

ance  x^  and  small  resistance  r^  so  that  —  is  very  small  and  —  is 

r\  r2 

very  large.  The  electromotive  force  Ev  Fig.  63,  consumed  in 

the  line  is  due  almost  wholly  to  resistance  and  if 
El  is  not  very  large  then  E2  is  very  nearly  equal  to 
E.  That  is  the  resistance  drop  in  a  transmission  line 


Fig.  63. 


Fig.  64. 


produces  but  very  little  diminution  of  electromotive  force  at  the 
terminals  of  a  receiving  circuit  of  large  reactance. 


APPLICATION   OF   THE   SYMBOLIC    METHOD.  87 


x  x 

4.  When  —  is  very  large  and  —  very  small  the  state  of  affairs 


is  shown  in  Fig.  64,  and  E  =  V  E*  -f  E22,  numerically. 

Example. — A  transmission  line  of  large -reactance  x^  and  small 
resistance  r^  supplies  current  to  a  receiving  circuit  of  large  resist- 
ance r2  and  small  reactance  x2.  The  electromotive  force  Ev 
Fig.  64,  consumed  in  the  line  is  due  almost  wholly  to  reactance 
and  if  El  is  not  very  large,  then  E2  is  very  nearly  equal  to  E. 
That  is,  the  reactance  drop  in  a  transmission  line  affects  the  elec- 
tromotive force  at  the  terminals  of  a  non-inductive  receiving  cir- 
cuit but  little. 

5.  When  —  is  large  and  positive  and  —  is  large,  but  negative, 

r\  r2 

then  jE'j  is  nearly  90°  ahead  of  /  and  £2  is  nearly  90°  behind  7, 

as  shown  in  Fig.  65.     The  figure  shows  the  limiting  case  for 

which  —  =  -f  oo  and  —  =  —  oo .     In  this  case  the  total  electro- 

ri  r2 

motive  force  E  is  numerically  equal       j 

to  the  difference  between  E,  and  E,.. 

j^ 

Examples. — (a)  A  transmission  line 
of  small  resistance  rv  and  large  react-        , 
ance  xl  supplies  current  to  a  conden-     ^ 
ser.     The  state  of  affairs  is  shown  in 
Fig.  65  and  the  electromotive  force      <« 

fe^ 

E.-,  at  the  terminals  of  the  condenser 

exceeds  the  generator  electromotive 

force  E  by  the  amount  Ev    Therefore 

reactance  drop  in  a  transmission  line 

increases  the  electromotive  force  at  the  terminals  of  a  receiving 

circuit  of  negative  reactance. 

(ft)  A  transmission  line  of  small  resistance  rl  and  large  react- 
ance xl  supplies  current  to  a  synchronous  motor  running  at  light 
load,  electromotive  force  of  motor  being  greater  than  electromotive 
force  of  generator.  In  this  case  the  electromotive  force  at  the 
terminals  of  the  receiving  circuit  is  nearly  90°  behind  the  current 


T. 


Fig.  65. 


88 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


in  phase,  as  in  case  of  the  condenser,  and  the  electromotive  force 
at  the  receiving  circuit  terminals  is  increased  by  the  reactance 
drop  in  the  line. 

(c)  An  inductance  is  connected  in  series  with  a  condenser  be- 
tween alternating  current  mains.  If  the  resistance  of  the  circuit 
is  comparatively  small  the  electromotive  force  at  the  condenser 
terminals  is  nearly  equal  to  the  sum  of  the  electromotive  force 
between  mains  plus  the  electromotive  force  between  the  terminals 
of  the  inductance.  (See  Articles  56  and  57.) 

64.  Problem  VIII.  Coils  in  parallel. — An  alternating  current 
/  divides  between  two  coils  connected  in  parallel  as  shown  in 
Fig.  66.  It  is  required  to  find  7X  and  72  each  in  terms  of  /,  rv 
rv  x^  and  xy  The  general  relation  between  /,  Iv  72  and  E  is 
shown  in  Fig.  67.  This  figure  is  given  in  order  that  the  follow- 
ing equations  may  be  more  easily  understood.  The  problem 
under  discussion  is  greatly  simplified  if  we  use  conductance  g  and 
susceptance  b  of  each  circuit  instead  of  resistance  and  reactance. 
According  to  the  definitions  given  in  Article  61  we  have  : 


(a) 


From  equation  (52)  we  have  : 


and 


Furthermore  the  total  current  /  is  the  vector  sum. of  II  and  /2, 
so  that : 

/=  7,  +  7,  (rf) 


APPLICATION   OF   THE   SYMBOLIC    METHOD.  89 


^     ' 


Fig.  66. 


Fig.  67. 


This  is,  of  course,  a  complex  equation.  Substituting  the 
values  of  7j  and  72  from  (£)  and  (<r)  in  (d)  and  solving  for  £ 
we  have 


-/&.+.« 

Substituting  this  value  of  E  in  (&)  and  (r)  we  have : 


and 


L  = 


These  equations  (/)  and  (g)  express  the  currents  1^  and  72  in 
terms  of  the  known  quantities,  7,  gv  gy  bl  and  bT  For  purposes 
of  numerical  calculations  the  components  of  each  7t  and  72  must 
be  separated.  The  numerical  value  of  each  is  then  found  by  tak- 
ing the  square  root  of  the  sum  of  the  squares  of  its  components. 

The  following  simple  cases  of  the  problem  under  consideration 
are  interesting. 

1.  When  x^  —  o  and  x^  —  o  ;  then  7,  7T,  72  and  E  are  parallel 
and  7=  7t  +  72  (numerically). 

2.  When  — !  =  —  then  7,  7p  and  72  are  parallel  to  each  other 

r\       rz 
and  all  behind  E  in  phase  by  the  angle  of  which  the  tangent  is 

— •     In  this  case,  also,  7=  7t  -f  72  (numerically). 


9o 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


3.  When  —  is  very  small  and  —  is  veiy  large  (or  vice  versa).     In 

this  case  7t  is  parallel  to  E  and  72  is  90°  behind  E  (or  vice  versa) 

as  shown  in  Figs.  68  and  69.     These 

+•     figures  are  particular  cases  of  Fig.  67. 

In  the  present  case  the  numerical  re- 
lation between  /,   II  and    /2  is   /  = 


Fig.  68. 


Fig.  69. 


-f  722  and  when  either  7t  or  /2  is  small  the  other  is  sensibly 
equal  (numerically)  to  /. 

x  x 

4.  When  -1  is  large  and  positive  and  —  is  large  but  nega- 

r\  rz 

tive,  then  fl  is  nearly  90°  behind  E,  and  /2  is  nearly  90°  ahead 

of  E,  as  shown  in  Fig.  70.     The  figure  shows  the  limiting  case 
for  which  —  =  -4-  oo  and  —  =  —  oo.      In  this  case  /=  /„  —  /., 

Y  Y  1.1' 

r\  T1 

numerically. 

Examples  of  case  </.  —  A  condenser  and  an  inductance  are  con- 
nected in  parallel  in  an  alternating  current  circuit.     The  current 

1  in  the  circuit  divides  in  the  two 
branches  formed  by  the  inductance  and 
the  condenser.  The  currents  7j  and 
/2  in  these  two  branches  are  nearly  op- 
posite to  each  other  in  phase  and  the 
».  current  /  is  numerically  equal  to  the 
difference  of  7}  and  /2. 

Il    the  frequency  of  the  alternating 


E 


Fig.  70. 


current  /  is  such  that  &L  and 


are 


equal  then  II  and  72  will  be  nearly  equal  ;  and 
be  very  much  greater  than  /. 


and  /2  will  each 


APPLICATION   OF   THE   SYMBOLIC   METHOD. 


65.  Compensation  for  lagging  current  in  a  receiving  circuit. — 
A  transmission  line  delivers  a  lagging  current  72  to  a  receiving 


2 

as 


circuit  of  resistance  r, 
and  reactance  ;r2 
shown  in  Figs.  71  and 
72.  A  condenser  con- 
nected in  parallel  with 
the  receiving  circuit 
takes  current  Iv  which 
is  90°  ahead  in  phase 
of  the  electromotive 
force  E  which  acts  upon 
the  receiving  circuit. 


transmission  line 


' 


transmission  line 

Fig.  71. 

The  total  current,  /,  generated  by  the  dis- 
tant alternator  and  delivered  by  the  transmission  line  may  be 
reduced  in  value  and  brought  into  phase  with  E,  if  the  reactance 

of  the  condenser  is  so  chosen 
that  the  current  taken  by  the 
condenser  is  equal  and  op- 
posite to  the  component  of  72 
which  is  at  right  angles  to  E. 
Discussion. — The  current 
Fig.  72.  /2  is,  by  equation  (5 1), 


or 


',  + 


»-,*  + 


so  that  the  component  of  /2  perpendicular  to  E  is  —  /• 


2      ,  2' 

m       H^    **-<, 


and,  to  bring  the  resultant  current  /  into  phase  with  E,  as  shown 
in  Fig.  72,  this  component  of  /2  must  be  equal  and  opposite  to 
the  current  Ejjxv  which  flows  into  the  condenser.  Therefore 


92 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


or 


which  expresses  the  value  of  the  negative  reactance  of  the  con- 
denser, in  order  that  the  current  delivered  by  the  transmission 
lines  may  be  in  phase  with  the  electromotive  force  acting  on  the 
receiving  circuit. 

Remark  i . — This  arrangement  is  seldom  used  in  practice,  on 
account  of  the  large  and  expensive  condensers  required  and  on 
account  of  the  very  considerable  loss  of  power  in  such  large 
condensers.  See  problem  65. 

Remark  2. — It  is  shown  in  Chapter  XII.  that  the  synchronous 
motor  running  at  light  load  has  negative  reactance,  if  its  electromo- 
tive force  is  greater  than  the  electromotive  force  of  the  generating 
alternator.  Such  a  synchronous  motor,  called  an  over-excited  syn- 
chronous motor,  may  be  used  to  compensate  for  lagging  currents. 

66.  The  transformer  without  iron. — Two  coils  of  wire  A  and  B, 
Fig-  73>  are  placed  near  together,  but  not  electrically  connected. 
The  coil  A,  called  the  primary  coil,  receives  alternating  current 
from  a  generator ;  its  resistance  is  r^  and  its  inductance  is  Lr 
The  coil  B,  called  the  secondary  coil,  is  connected  to  a  receiving 

circuit.  The  total  resistance 
of  coil  B  and  its  receiving  cir- 
cuit is  r2,  and  the  total  induc- 
jlf]  '  *  Vo  tance  is  LT  It  is  required  to 
find  the  electromotive  force 
which  must  act  upon  A  to 
maintain  in  it  a  given  harmonic 
current  7r  To  solve  this 
problem,  one  must  consider 

the  electromotive  force  induced  in  each  coil  by  the  changing  cur- 
rent in  the  other  coil.  This  electromotive  force  in  one  -:oil  is 
proportional  to  the  rate  of  change  of  the  current  in  the  other 
coil,  and  the  proportionality  factor  J/has  a  definite  mutual  value 
for  the  two  coils. 


APPLICATION   OF   THE   SYMBOLIC    METHOD.  93 

Determination  of  the  secondary  current  72.  —  The  given  primary 
current  induces  in  the  secondary  coil  an  electromotive  force 

which  is  at  each  instant  equal  to  M  -^  - 

This  electromotive  force  is  90°  ahead  of  /p  its  effective  value 
is  a)MIv  and  its  symbolic  expression  is  j»Affv  and  according  to 
equation  (51)  this  electromotive  force  produces  in  the  secondary 
coil  a  current 


/,-  \  (a) 

2 


Reaction  of  the  secondary  current  upon  the  primary  coil.  —  The 
secondary  current  /2  induces  in  the  primary  coil  an  electromo- 

tive force  which  is  at  each  instant  equal  to  M  —^  •    This  electro- 

motive force  is  90°  ahead  of  72  in  phase,  its  effective  value  is 
<»MIV  and  its  complex  expression  is  jo)MI2.  This  electromo- 
tive force  induced  in  the  primary  by  the  secondary  current  must 
be  overcome  by  the  electromotive  force  which  acts  upon  the  pri- 
mary. The  portion  of  the  acting  electromotive  force  which  thus 
balances  the  reaction  of  the  secondary  current  is  equal  to  this 
reaction  and  opposite  to  it  in  sign  and  is,  therefore,  equal  to 
->J//2. 

Determination  of  total  electromotive  force  acting  on  primary.  — 
This  total  electromotive  force  consists  of  three  parts  as  follows  : 

I.  The  part  described  above  which  balances  the  reaction  of 
the  secondary  current.  This  part  is  equal  to  —ja>Mf2  or  using 
the  value  of  72  from  equation  (a)  we  have  for  this  part  of  the 
total  electromotive  force 


2.  The  part  used  to  overcome  the  resistance  of  the  primary 
coil.     This  is  at  each  instant  equal  to  rjv  its  effective  value  is 
rjv  and  its  complex  expression  is  rj^  since  it  is  in  phase  with  Ir 

3.  The  part  used  to  overcome  the  inductance  of  the  primary 


94 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


coil.     This  is  at  each  instant  equal  to  L^  ~,  it  is  90°  ahead  of 

7j,  its  effective  value  is  <»>LJV  and  its  complex  expression  is 
jo)LlIr  Therefore  the  total  electromotive  force  required  to 
maintain  the  given  primary  current  is 


or  separating  components 


Fig.  74  shows  the  primary  current 
/,  the  electromotive  force  jtoMIl  induced 
in  the  secondary  coil,  the  secondary 
current  /2,  the  portion  of  the  primary 
electromotive  force  —  /o>J//2  used  to 
balance  the  reaction  of  the  secondary 
current,  the  portion  of  the  primary 
electromotive  force  r^Il  used  to  over- 
come primary  resistance,  and  the  portion 
of  the  primary  electromotive  force 
/wZj/j  used  to  overcome  primary  in- 
ductance. The  total  primary  electro- 
motive force  is  the  vector  sum  of 


Fig.  74. 


Equation  (c}  shows  that  the  effect  of  the  secondary  coil  is  to 
make  the  primary  coil  behave  as  if  its  resistance  were  increased 
by  the  amount 


and  its  inductance  decreased  by  the  amount 


67.  Inductance  error  of  the  wattmeter. — It  was  shown  in  Article  41  that  a  watt- 
meter which  has  been  calibrated  with  continuous  electromotive  force  and  current  indi- 


APPLICATION   OF   THE   SYMBOLIC    METHOD.  95 

cates  power  correctly  when  used  with  alternating  current,  provided  the  inductance  of 
the  shunt  coil  is  zero.  When  the  shunt  circuit  has  inductance,  however,  the  watt- 
meter indicates  incorrectly  when  used  with  alternating  currents.  The  following  dis- 
cussion refers  to  Fig.  27,  and  it  is  assumed  that  the  wattmeter  is  provided  with  a 
Weston  compensating  coil  so  that  the  current  in  coil  6,  Fig.  27,  may  be  assumed  to  be 
equal  to  the  current  in  the  receiving  circuit  CC. 

Let  R  be  the  resistance  of  the  receiving  circuit  and  X  its  reactance  ;  and  let  r  be 
the  resistance  of  the  shunt  circuit  and  x  its  reactance. 

Imagine  the  wattmeter  connected  to  direct  current  mains  as  shown  in  Fig.  27  so 
that  a  continuous  current  C  flows  through  the  coil  b  and  the  receiving  circuit.  The 
electromotive  force  between  the  terminals  of  the  receiving  circuit  is  CR,  and  the  cur- 
rent in  the  shunt  circuit  is  CR\r.  The  force  action  between  the  wattmeter  coils  is 
proportional  to  the  product  of  the  two  currents  C  and  CR\r,  or  to  C*R\r  ;  and  C*R, 
the  watts  expended  in  the  receiving  circuit,  is  the  wattmeter  reading  W. 

Imagine  the  wattmeter  connected  to  alternating  current  mains,  as  shown  in  Fig.  27, 
so  that  an  alternating  current  /  flows  through  the  coil  b  and  the  receiving  circuit.  The 
electromotive  force  between  the  terminals  of  the  receiving  circuit  is  I(R-\-jX),  and 

the  current  in  the  shunt  circuit  is  —  --  •   The  component  of  this  current  parallel 

to  I  is     ^    ^~       —  and  the  product  of  this  component  into  /  gives  the  average  prod- 

uct *  of  the  two  currents,  viz  :  —  2  —  -  —  -  •  If  this  alternating  current  gives  the 
same  deflection  of  the  wattmeter  as  the  above-mentioned  direct  current  we  have 


since  equal  deflections  require  equal  force  actions,  and  equal  force  actions  depend  upon 
equal  current-products. 

Multiply  both  members  of  (i)  by  fi,  write  fF(  wattmeter  reading)  for  C2R,  and 
write  P  for  I*R,  this  being  the  watts  actually  expended  in  the  receiving  circuit  by  the 
alternating  current,  and  we  have,  after  solving  for  P, 


~  r(Rr-Xx) 

That  is,  the  wattmeter  reading  Wmust  be  multiplied  by          --  ^r4r  to  give  the 

f  {Rr— 

power  expended  in  the  receiving  circuit. 


PROBLEMS. 

63.  A  transmission  line  having  an  inductance  of  0.02  henry 
and  a  resistance  of  25  ohms  supplies  current  at  60  cycles  per 

*  Compare  Article  52. 


g6  ELEMENTS   OF   ALTERNATING   CURRENTS. 

second  to  a  condenser  of  which  the  capacity  is  24  microfarads. 
The  generator  electromotive  force  is  20,000  volts.  What  is  the 
electromotive  force  at  the  condenser  terminals?  Ans.  20,870 
volts. 

64.  A  circuit  a  consists  of  50  ohms  resistance  and  2  henrys 
inductance  ;  another  circuit  b  consists  of  50  ohms  resistance  and 
a  0.75  microfarad  condenser.     The  two  circuits  a  and  b  are  con- 
nected in  parallel  in  a  circuit  in  which  a  125-cycle  current  of  I 
ampere  is  flowing.     What  is  the  current  in  a  and  what  is  the 
current  in  £?     Ans.   Current  in  a=  12.02  amperes,  current  in 
b  =  11.13  amperes. 

65.  An  alternator  delivers  100  amperes  at  1,100  volts  and  60 
cycles  to  a  receiving  circuit  of  which  the  power  factor  is  0.85. 
What  capacity  condenser  would  be  required  to  compensate  for 
lagging  current  ?     What  number  of  leaves  of  a  paraffined  paper 
condenser  20  by  25  centimeters  would  be  required  for  this  con- 
denser, thickness  of  paraffined    paper  being   0.08   cm.?     Take 
inductivity  of  paraffined  paper  equal  to  2.     Ans.    127.1   micro- 
farads, 114,875  leaves. 

66.  A  transformer  (without  iron)  consists  of  two  long  cylin- 
drical coils  each  having  10  turns  of  wire  per  centimeter  of  length 
(one  layer).     The  coils  are  each  50  cm.  long  and  their  radii  are  2 
cm.  and  3   cm.   respectively,  the  smaller  coil  being  inside  the 
larger.      Calculate  the  value  and  phase  of  the  electromotive  force 
required  to  maintain  a  current  of  10  amperes  at  60  cycles  per 
second  in  the  outer  coil  ;  calculate  the  current  in  the  inner  coil, 
and  calculate  the  true  and  apparent  reactance  and  apparent  re- 
sistance of  the  outer  coil.     The  outer  coil  has  2  ohms  resistance 
and  the  inner  coil  has  I  y2  ohms  resistance  and  its  terminals  are 
short-circuited.*     Ans.    21.123    volts,  6°2i'    phase    difference, 

*  The  mutual  inductance,  in  henrys,  of  two  coaxial  solenoids  is,  approximately, 

io9 


in  which  zf  and  z"  are  the  turns  of  wire  per  unit  length  on  the  respective  coils,  rfr  is 
the  radius  of  the  inside  coil,  and  /  is  the  length  of  the  coils. 


APPLICATION   OF   THE   SYMBOLIC    METHOD.  97 

1.  5  3  84  amperes,  0.66967  true  reactance,  0.66896  ohms  reactance, 
2.0036  ohms  resistance. 

67.  The  expression  (V),  Article  66,  may  be  written 


where  R  is  the  apparent  change  of  resistance  of  the  primary  coil 
due  to  the  presence  of  the  secondary  coil  and  X  is  the  apparent 
change  of  reactance  of  the  primary  coil  due  to  the  presence  of 
the  secondary  coil.  The  primary  coil  has  in  circuit  with  it  an 
adjustable  non-inductive  resistance  so  that  rl  may  be  varied  at 
will.  Assume  that  R  and  X  are  small  compared  to  rv  and  ^  so 
that  R2  and  X2  are  negligible.  Show  that  the  impedance 
(numerical  value)  of  the  primary  circuit  is  not  altered  by  the 
presence  of  the  secondaiy  when  rj^  =  R/X.  How  is  the  im- 
pedance (numerical  value)  of  the  primary  circuit  affected  by  the 
presence  of  the  secondary  when  rl  is  less  than  ^R/X  ;  when  r^ 
is  greater  than  ^R/X? 

68.  A  wattmeter  is  to  be  used  on  a  circuit  of  which  the  power 
factor  is  0.85.  The  resistance  of  the  shunt  circuit  is  1,000  ohms. 
Find  the  four  values  of  reactance  of  shunt  circuit  for  which  the 
true  power  will  differ  from  the  reading  by  ±  0.005  °f  tne  read- 
ing. Ans.  Reading  too  great  x  =  -f  7.45  or—  631.05,  read- 
ing too  small  x  —  —  8.  1  5  or  —  608.75. 


CHAPTER   VIII. 
SINGLE-PHASE   AND   POLYPHASE   ALTERNATORS. 

68.  The  single-phase  alternator  and  its  limitations. — The  simple 
alternator  described  in  Chapter  II.  is  called  the  single-phase  alter- 
nator.    It  has  one  pair  of  collector  rings  to  which  the  terminals 
of  the  armature  windings  are  connected.     The  current  given  by 
the  single-phase  alternator  is  entirely  satisfactory  for  the  opera- 
tion of  electric  glow  lamps,  fairly  satisfactory  for  the  operation 
of  electric  arc  lamps,  and  in  general  for  all  purposes  in  which 
the    heating    effect,    only,    of    the    current    is   important.      For 
motive  purposes  the  simple  alternating  current  is  not  satisfactory, 
as  it  is  difficult  to  make  a  single-phase  alternating-current  motor 
which  will  start  satisfactorily  under  load.     For  electrochemical 
processes  the  alternating  current  cannot  be  used.     The  satisfac- 
tory use  of  alternating    currents  for  motive    purposes  depends 
mainly  upon  the  use  of  the  induction  motor  described  in  Chapter 
XIII.     It  is  the  requirements  of  this  motor  which  have  led  to  the 
development  of  the  polyphase  systems. 

69.  The  two-phase  alternator. — Consider  two  similar  and  inde- 
pendent single-phase  armatures  A  and  B,  Fig.  75,  mounted  rig- 
idly on  the  same  shaft,  one  beside  the  other,  and  revolving  inside 
the  same  crown  of  field  magnet  poles.      In  the  figure,  armature 
B  is  shown  inside  of  A  for  the  sake  of  clearness.     These  arma- 
tures are  so  mounted  on  the  shaft  that  the  slots  of  A  are  midway 
under  the  poles  when  the  slots  of  B  are  midway  between  the  poles 
as  shown.     Under  these  conditions  the  electromotive  forces  of  A 
and  B  are  so  related  in  their  pulsations  that  the  electromotive 
force  of  A  is   at  its  maximum  when  the  electromotive  force  of 

98 


SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS.        99 


B  is  zero,  that  is,  the  electromotive  forces  are  90°  apart  in  phase, 
or  in  quadrature  with  each  other.  Two  alternators  connected 
(mechanically)  in  the  manner 
indicated  constitute  a  two- 
phase  alternator.  The  two 
distinct  and  independent  elec- 
tromotive forces  generated  by 
such  a  machine  are  used  to 
supply  two  distinct  and  inde- 
pendent currents  to  two  dis- 
tinct and  independent  circuits. 
In  practice  the  two-phase 
alternator  is  made  by  placing 
the  armature  windings  of  A 
and  B  upon  one  and  the 

same  armature  body.     For  this  purpose  the  armature  body  has 
twice  as  many  slots  as  A  or  B,  Fig.   75.     Fig.   76  shows  such 

an  armature.  The 
slots  marked  av  a2, 
ay  etc.,  receive  the 
conductors  belong- 
ing to  phase  A,  and 
those  marked  bv  bv 
by  etc.,  receive  those 
belonging  to  phase 
B.  The  A  winding 
would  pass  up*  slot 
alt  down  a.2,  up  a^  and 
so  on,  the  terminals 
of  the  winding  being 
connected  to  two  col- 
lector rings.  The  B 
Fig.  76.  winding  would  pass 

up  slot  bv  down  bv  up  by  and  so  on,  its  terminals  being  connected 

*  Up  and  down  being  parallel  to  the  armature  shaft  to  and  from  one  end  of  the 
armature. 


100 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


to  two  collector  rings  distinct  from  those  to  which  the  A  winding 
is  connected. 

The  armature  windings  A  and  B  here  described  are  of  the  con- 
centrated type  (see  Article  20),  having  only  one  slot  per  pole  for 
each  winding.  Distributed  windings  also  are  frequently  used  for 
two-phase  alternators.  Thus  Fig.  77  shows  a  portion  of  a  two- 
phase  armature  with  its  A  and  B  windings  each  distributed  in  two 


Fig.  77. 


slots  per  pole.  The  coils  belonging  to  windings  A  and  B  are 
differently  shaded  to  distinguish  them.  The  coils  belonging  to 
winding  A  are  connected  together  in  a  manner  indicated  by  the 
dotted  lines  and  the  coils  belonging  to  winding  B  are  connected 
together  as  indicated  by  the  full  lines.  (See  Article  84.) 

Two-phase  alternators  are  usually  provided  with  two  sets  of 
collector  rings  ;  one  ring  may,  however,  be  made  to  serve  as  a 
common  connection  for  the  two  armature  windings,  as  shown  in 


Fig.  78. 


SINGLE-PHASE  AND   POLYPHASE*  *  A^TERNATO  RS>  '  !<>*' 


Fig.  78.  In  this  case  a  three-wire  transmission  line  is  used  and 
the  separate  receiving  circuits  x  and  y  are  connected  as  shown  in 
the  figure.  The  objection  to  this  arrangement  is  that  when  the 
receiving  circuits  x  and  y  are  inductive  the  electromotive  forces 
at  the  terminals  of  x  and  y  respectively  are  not  equal  and  not  90° 
apart  in  phase  on  account  of  the  electromotive  force  lost  in  the 
common  main  3.  This  disarrangement 
of  the  electromotive  forces  in  a  polyphase 
system  is  called  distortion. 


70.  Two-phase    electromotive    forces 
and  currents. — The  two  lines  A  and  B, 
Fig.    79,    represent    the    electromotive 
forces  of  the  A  and  B  windings  respec- 
tively of  a  two-phase  alternator.      If  the  B 
circuit  which  receives  current  from  A  is                       ^^-^ 
of  the  same  resistance  and  reactance  as 

the  circuit  which  receives  current  from  B  then  the  system  is  said 
to  be  balanced  and  each  current  lags  behind  its  electromotive  force 
by  the  same  amount.  In  this  case  the  currents  are  equal  and  in 
quadrature  with  each  other  and  are  represented  by  the  two  dotted 
lines  a  and  b  in  Fig.  79. 

71.  Electromotive  force  and  current  relations  in  two-phase  three-wire  system. 
Electromotive  force. — The  electromotive  force  between  the  mains  I  and  2,  Fig.  78,  is 
the  vector  sum  *  of  the  electromotive  forces  A_aad.jB,  Fig.  79.     This  electromotive 
force  is  therefore  represented  by  the  diagonal  of  the  parallelogram  constructed  on  A 
and  B,  Fig.  79.     It  is  45°  behind  A  in  phase  and  its  effective  value  is  V  2.E  where 
E  is  the  common  effective  value  of  the  electromotive  forces  A  and  B. 

Current. — The  current  in  main  3  is  the  vector  sum*  of  the  currents  in  mains  I 
and  2,  namely  a  and  6,  Fig.  79.  This  current  is  therefore  represented  by  the  diag- 
onal of  the  parallelogram  constructed  on  a  and  b,  Fig.  79;  it  is  45°  behind  a  in 
phase  and  its  effective  value  is  V 2! where  /is  the  common  effective  value  of  the  cur- 
rents a  and  b. 

Effect  of  line  drop  in  the  common  main  of  a  two-phase  three-wire 
system. — A  two-phase  alternator  supplies  current  over  three  wires 
to  two  similar  inductive  receiving  circuits  x  and  y  as  shown  in 

*Or  difference  according  to  convention  as  to  signs. 


:  CLEMENTS  OF  ALTERNATING  CURRENTS. 


Fig.  78.  The  lines  A  and  B,  Fig.  80,  represent  the  generator 
electromotive  forces,  a  represents  the  current  in  main  I,  b  repre- 
sents the  current  in  main  2,  and  c  represents  the  current  in  the 
common  main  3.  The  lines  Re,  parallel  to  r,  represent  the  elec- 


Fig.  80. 

tromotive  force  lost  in  main  3,  the  line  (9  to  x  represents  the 
electromotive  force  between  the  terminals  of  receiving  circuit  x, 
and  the  line  (9  to  y  represents  the  electromotive  force  between 
the  terminals  of  receiving  circuit  y.  The  electromotive  forces 
lost  in  mains  I  and  2  are  not  considered  in  this  discussion  inas- 
much as  they  do  not  tend  to  distort  the  two-phase  electromotive 
forces. 

72.  The  three-phase  alternator. — Consider  three  similar  single- 
phase  armatures,  A,  B  and  C,  mounted  side  by  side  on  the  same 
shaft  and  revolved  in  the  same  field,  each  armature  having  as 
many  slots  as  there  are  field  poles.  Fix  the  attention  upon  a 
certain  armature  slot  of  A  and  let  time  be  reckoned  from  the  in- 
stant that  this  slot  is  squarely  under  an  TV-pole.  Let  t  be  the  time 
which  elapses  as  this  armature  slot  passes  from  the  center  of  one 
TV-pole  to  the  center  of  the  next  TV-pole.  The  armature  B  is  to 
be  so  fixed  to  the  shaft  that  its  slots  are  squarely  under  the  poles 
at  the  instant  ^/,  and  the  armature  C  is  to  be  so  fixed  that  its 
slots  are  squarely  under  the  poles  at  the  instant  y^t.  While  a 
slot  passes  from  the  center  of  one  TV-pole  to  the  center  of  the 


SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS.       103 


next  zV-pole  the  electromotive  force  passes  through  one  complete 
cycle.  Hence  the  electromotive  forces  given  by  three  armatures, 
arranged  as  above,  will  be 
120°  apart  in  phase,  as 
shown  in  Fig.  81,  in 
which  the  lines  A,  B  and 
C  represent  the  respec- 
tive electromotive  forces. 
The  currents  given  by 
the  armatures  to  three 
similar  receiving  circuits 
lag  equally  behind  the 
respective  electromotive 
forces  and  are  represented 
by  the  dotted  lines  a,  b  Fi§'81- 

and  c.    This  combination  of  three  alternators  is  called  a  three-phase 
alternator.     In  practice  the  three  distinct  windings  A,  B  and  C  are 


&___ 


Fig.  82. 


placed  upon  one  and  the  same  armature  body.     For  this  purpose 
the  armature  body  has  three  times  as  many  slots  as  A,  B  or  C. 


104 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


Fig.  82  shows  the  arrangement  of  the  slots  for  such  a  winding. 
The  slots  belonging  to  phase  A  are  drawn  in  heavy  lines  and  are 
marked  av  a.2,  etc.  Those  belonging  to  phase  B  are  shown 
dotted  and  those  belonging  to  phase  C  are  shown  in  light  lines. 
The  A  winding  would  pass  up  slot  av  down  av  up  #3,  etc.;  the 
B  winding,  up  bv  down  bv  up  by  etc.;  and  similarly  for  phase  C. 


Fig.  83. 


The  windings  A,  B  and  C  here  described  are  of  the  concen- 
trated type,  having  only  one  slot  per  pole  for  each  winding.  Dis- 
tributed windings  also  are  frequently  used  for  three-phase  alter- 


Fig.  84. 

nators.  Thus  Fig.  83  shows  a  portion  of  a  three-phase  armature 
with  its  Ay  B  and  C  windings  each  distributed  in  two  slots  per 
pole.  The  coils  belonging  to  windings  A,  B  and  C  respectively 
are  differently  shaded  to  distinguish  them.  The  manner  of  con- 
necting the  coils  of  each  winding  is  described  in  Article  84. 
If  the  three  circuits  of  a  three-phase  alternator  are  to  be  en- 


SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS.       105 


Fig.  85. 


tirely  independent  six  collector  rings  must  be  used,  two  for  each 
winding  ;  however,  the  circuits  may  be  kept  practically  independ- 
ent by  using  four  collector 
rings  and  four  mains,  as 
shown  in  Fig.  84. .  The 
main  4  serves  as  a  com- 
mon return  wire  for  the 
independent  currents  in 
mains  I,  2  and  3.  When 
the  three  receiving  circuits 
are  equal  in  resistance  and 
reactance,  that  is,  when 
the  system  is  balanced,  the 
three  currents  are  equal 
and  120°  apart  in  phase 
(each  current  lagging  be- 
hind its  electromotive  force  by  the  same  amount)  and  their  sum 
is  at  each  instant  equal  to  zero  :  in  which  case  main  4,  Fig.  84, 
carries  no  current  and  this  main  and  the  corresponding  collector 
ring  may  be  dispensed  with,  the  three  windings  being  connected 
together  at  the  point  N,  called  the  common  junction.  This  ar- 
rangement, shown  in  the 
symmetrical  diagram,  Fig. 
85,  is  called  the  Y,  or  star, 
scheme  of  connecting  the 
three  windings  A,  B  and  C. 
Another  scheme  for  con- 
necting the  three  windings 
A,  B  and  C  (also  for  bal- 
anced loads)  called  the  A 
(delta)  or  mesh  scheme  is 
shown  in  Fig.  86.  Winding 
A  is  connected  between 
rings  3  and  I,  winding  B 
between  rings  I  and  2  and  winding  C  between  rings  2  and  3. 


Fig.  86. 


io6 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


The  direction  in  a  circuit  in  which  the  electromotive  force  or 
current  is  considered  as  a  positive  electromotive  force  or  current 
is  called  the  positive  direction  through  the  circuit.  This  direction 
is  chosen  arbitrarily.  The  arrows  in  Figs.  85  and  86  indicate 
the  positive  directions  in  the  mains  and  through  the  windings. 
It  must  be  remembered  that  these  arrows  do  not  represent  the 
actual  directions  of  the  electromotive  forces  or  currents  at  any 
given  instant,  but  merely  the  directions  of  positive  electromotive 
forces  or  currents.  Thus  in  Fig.  85  the  currents  are  considered 
positive  when  flowing  from  the  common  junction  towards  the 
collecting  rings  and  the  currents  are  never  all  of  the  same  sign. 

73.  Electromotive  force  and  current  relations  in  Y-connected 
armatures.  Electromotive  force  relations. — Passing  through  the 


Fig.  87. 


windings  A  and  B  from   ring   2    to  ring    i,*  in    Fig.  85,  the 
winding  A  is  passed  through  in  the  positive  direction  and  the 

*  Which  is  the  direction  in  which  an  electromotive  force  must  be  generated  to  give 
an  electromotive  force,  acting  upon  a  receiving  circuit  from  main  I  to  main  2. 


SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS.       IO/ 


winding  B  in  the  negative  direction.  Therefore  the  electro- 
motive force  between  mains  I  and  2  is  A  —  B.  Similarly  the 
electromotive  force  between  mains  2  and  3  is  B  —  C  and  the 
electromotive  force  between  mains  3  and  I  is  C—  A.  These 
differences  are  shown  in  Fig.  87.  The  electromotive  force  be- 
tween mains  I  and  2,  namely,  A  —  B,  is  30°  behind  A  in  phase 
and  its  effective  value  is  2E  cos  30°  —  */$&,  where  £  is  the  com- 
mon value  of  each  of  the  electromotive  forces  A,  B  and  C.  Sim- 
ilar statements  hold  concerning  the  electromotive  forces  between 
mains  2  and  3  and  between  mains  3  and  I.  Hence  the  electro- 
motive force  between  any  pair  of  mains  leading  from  a  three- 
phase  alternator  with  a  Y-connected  armature  is  equal  to  the 
electromotive  force  generated  per  phase  multiplied  by  \/3« 

Current  relations. — In  the  Y  connection  the  currents  in  the 
mains  are  equal  to  the  currents  in  the  respective  windings,  as  is 
evident  from  Fig.  8.5. 

74.   Electromotive  force  and  current  relations  in  A-connected 
armatures.     Electromotive  force  relations. — In  A-connected  arma- 
tures    the     electromotive 
forces  between  the  mains 
or     collector     rings     are 
equal  to  the  electromotive 
forces    of    the    respective 
windings,    as    is    evident 
from  Fig.  86. 

Current  relations. — Re- 
ferring to  Fig.  86  we  see 
that  a  positive  current  in 
winding  A  produces  a 
positive  current  in  main  I 
and  that  a  negative  current 
in  winding  B  produces  a 

f.  .  Fig.  88. 

positive  current  in  main  I, 

therefore  the  current  in  main  I  is  a  -  b  when  a  is   the  current 


io8 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


main  i 


main  z 


B 

mmmm 


in  A  and  b  is  the  current  in  B.     Similarly  the  current  in  main 

2  is  b  —  c  and  the  current  in  main   3   is  c  —  a.     These    differ- 
ences are  shown  in  Fig.  88.     The  current  in  I,  namely  a-b,  is 
30°  behind  a  in  phase  and  its  effective  value  is  \/3  /  when  /  is 
the  common  effective  value  of  the  currents  a,  b,  c  in  the  different 
phases.     Similar  statements  hold  for  the  currents  in  mains  2  and 

3  ;  so  that  the  current  in  each  main  of  a  A-conrtected  armature 
is  \/3  times  the  current  in  each  winding. 

75.  Connection  of  receiving  circuits  to  three-phase  mains.  Dis- 
similar circuits  (unbalanced  system}. — When  the  receiving  circuits 
which  take  current  from  three-phase  mains  are  dissimilar,  four 

mains  should  be  employed 
as  indicated  in  Fig.  84 ; 
each  receiving  circuit  being 
connected  from  main  4  to 
one  of  the  other  mains.  It 
is,  however,  desirable  to 
keep  the  three  windings  A, 
B  and  C  of  the  alternator 
as  nearly  equally  loaded  as 
possible,  and  the  receiving 
circuits  are  so  disposed  in  practice  as  to  satisfy  this  condition  as 
nearly  as  possible. 

Similar  circuits  (balanced  system}. — When  three-phase  currents 
are  used  to  drive  induction  motors,  synchronous  motors  or  rotary 
converters,  each  unit  takes 
current  equally  from  the  three 
mains,  and  since  three-phase 
currents  are  utilized  mainly  in 
the  operation  of  the  machines 
mentioned,  the  system  is  usu- 
ally balanced.  In  this  case  . 
three  mains  only  are  employed 
and  each  receiving  unit  has  three  similar  receiving  circuits  connected 


mains 


Fig.  89. 


main  i 


main  z 


wains 


Fig.  90. 


SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS.       109 

to  the  mains  according  to  either  the  Y  or  A  method.  The  Y  method 
of  connecting  receiving  circuits  is  shown  in  Fig.  89.  One  terminal 
of  each  receiving  circuit  is  connected  to  a  main  and  the  other  ter- 
minals are  connected  together  at  IV.  In  this  case  the  current  in 
each  receiving  circuit  is  equal  to  the  current  in  the  main  to  which 
it  is  connected.  The  electromotive  force  between  the  terminals 

of  each  receiving  circuit  is  equal  to  — =  where  E  is  the  electro- 

^3 
motive  force  between  any  pair  of  mains. 

The  A  method  of  connecting  receiving  circuits  is  shown  in 
Fig.  90.  Here  the  three  receiving  circuits  are  connected  between 
the  respective  pairs  of  mains,  the  electromotive  force  acting  on 
each  receiving  circuit  is  the  electromotive  force  between  the 

mains,  and  the  current  in  each  receiving  circuit  is  —  =.  where  /  is 

^3 
the  current  in  each  main. 

76,  Power  in  polyphase  systems. — The  several  circuits  of  a 
polyphase  system  are  in  general  entirely  separate  and  independ- 
ent, and  the  total  power  delivered  to  a  receiving  apparatus  is  to 
be  found  by  measuring  the  power  delivered  to  each  separate  re- 
ceiving circuit ;  the  total  power  delivered  is  the  sum  of  the 
amounts  delivered  to  the  different  receiving  circuits. 

Balanced  systems. — When  a  polyphase  system  is  balanced  the 
several  circuits  are  entirely  similar  and  the  same  amount  of  power 
is  delivered  to  each  receiving  circuit  of  a  given  piece  of  receiving 
apparatus. 

Balanced  t'tvo-phase. — Let  E  be  the  electromotive  force  of  each 
phase,  /  the  current  furnished  to  each  of  two  similar  receiving 
circuits,  and  cos  6  the  power  factor  of  each  receiving  circuit. 
Then  El  cos  6  is  the  power  delivered  to  each  circuit,  so  that  the 

total  power  delivered  is  ,     , 

P=  2EI cos  d 

Balanced  three-phase. — Let  E  be  the  electromotive  force  be- 
tween the  terminals  of  each  receiving  circuit,  /  the  current  in  each 


110  ELEMENTS   OF   ALTERNATING   CURRENTS. 

receiving  circuit,  and  cos  6  the  power  factor  of  each  circuit. 
Then  El  cos  0  is  the  power  delivered  to  each  receiving  circuit,  so 
that  the  total  power  delivered  is 

P=$£Jcos0  (56) 

in  which,  as  must  be  remembered,  E  is  the  electromotive  force  at 
the  terminals  of  each  receiving  circuit  and  /  is  the  current  in  each 
receiving  circuit.  On  the  other  hand, 

P=\/~$EIcos6  (57) 

in  which  E  is  the  electromotive  force  between  each  pair  of  mains, 
/  is  the  current  in  each  main,  and  cos  0  is  the  power  factor  of 
each  receiving  circuit.  Equation  (57)  may  be  derived  from  (56) 
by  considering  that  the  current  Im  in  each  main,  for  the  case  of 
A  connection  for  example,  is  equal  to  v/3  7,  so  that,  substituting 

— ^for  /in  equation  (56)  we  have  equation  (57). 

77.  The  flow  of  energy  in  balanced  polyphase  systems. — It  was 
pointed  out  in  Article  26  that  the  power  developed  by  a  single- 
phase  alternator  pulsates  with  the  alternations  of  electromotive 
force  and  current.  The  power  delivered  to  a  balanced  system  by 
a  polyphase  generator,  on  the  other  hand,  is  not  subject  to  pul- 
sations, but  is  entirely  steady  and  constant  in  value. 

Discussion  for  a  two-phase  alternator. — Consider  a  single-phase  alternator  of  which 

the  electromotive  force  is 

e=  1$  sin  ut  (#) 

and  which  gives  a 'current 

*  =  Jsin  (ut  —  6) 
or 

i  =  I  sin  w/  cos  6  —  I  cos  ut  sin  6  (3) 

The  instantaneous  power  is 

ei=z  I$I  cos  6  sin2  ut— *•  I$I  sin  6  sin  «/  cos  ut 

which  pulsates  with  a  frequency  twice  as  great  as  the  frequency  of  e  and  i. 

Let  equations  (a)  and  (£)  express  the  electromotive  force  and  current  of  one  phase 
of  a  (balanced)  two-phase  alternator,  then  electromotive  force  and  current  of  the 
other  phase  are 

e'  =  J$  cos  uf  (c] 

i'  =  I  cos  (otf  — 0)  =  I  cos  ut  cos  0  +  J  sin  ut  sin  6  (d) 


SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS.      Ill 

The  instantaneous  power  output  of  this  phase  is 

efif  =  J$I  cos  6  cos2ut  -j-  J£ I  sin  6  sin  wt  cos  wt 
Therefore  the  total  power  output  of  the  two-phase  machine  is 
ft  -f-  cfi'  =  J$I  cos  6  ( sin2  ut  -j-  cos2  ut) 

which  is  constant. 

Remark. — The  torque  of  a  single-phase  alternator  pulsates 
with  the  pulsations  of  the  power  output.  In  a  balanced  poly- 
phase alternator,  however,  the  torque  is  steady,  since  the  power 
does  not  pulsate ;  also  polyphase  synchronous  motors,  rotary 
converters,  and  induction  motors  are  driven  by  a  steady  torque. 

78.  Measurement  of  power  in  polyphase  systems. — In  a  poly- 
phase system,  balanced  or  unbalanced,  the  power  taken  by  any 
unit,  such  as  an  induction  motor,  may  be  determined  by  measur- 
ing the  power  taken  by  each  single  receiving  circuit  and  adding 
the  results.  In  order  to  measure  the  power  taken  by  a  single 
receiving  circuit  the  current  coil  of  the  wattmeter  is  connected  in 
series  with  the  circuit  and  the  fine  wire  coil  is  connected  to  the 
terminals  of  the  circuit.  The  inconvenience  of  connecting  and 


main  a 


6 


Fig.  91. 


disconnecting  the  wattmeters  makes  it  necessary  to  use  a  separate 
wattmeter  for  each  receiving  circuit.  Two  wattmeters  are  suffi- 
cient for  the  complete  measurements  of  the  power  taken  by  any 
three-phase  receiving  unit.  The  connections  are  shown  in  Fig. 


112  ELEMENTS    OF   ALTERNATING   CURRENTS. 

91.     The  receiving  circuit  may  be  balanced  or  unbalanced  and 
connected  Y  or  A. 

Proof.  —  Let  the  positive  direction  in  the  mains  a  and  b  and  in  the  three  receiving 
circuits  be  chosen  as  indicated  by  the  arrows  in  Fig.  91.  These  directions  are  chosen 
symmetrically  with  respect  to  the  two  wattmeters.  Let  the  instantaneous  currents  in 
the  receiving  circuits  be  i't  if>  ',  and  i'"  as  shown.  Let  a  be  the  instantaneous  cur- 
rent in  main  a,  and  let  b  be  the  instantaneous  current  in  main  b.  Then 


The  reading  W  of  the  upper  wattmeter  is  equal  to  the  average  value  of  the  prod- 
uct of  the  current  a  which  flows  through  the  current  coil  of  the  instrument  into  the 
electromotive  force  ef  which  acts  upon  the  shunt  circuit  of  the  instrument.  That  is 

W  —  average  aef 
Similarly 

Wf?  =^  average  befr 

Substituting  the  above  values  of  a  and  b  in  the  expression  for  Wf  -J-  Wff  we  have 

W  -f  IV"  =  average  e'i'  -f-  average  e"i"  -\-  average  (  ef  —  e"  )  i'" 
But  ef  —  e"  =  e'"  so  that 

W  -f-  Wf/  =  average  e'i'  -f-  average  e"i"  -\-  average  e/f/i'ff 

Q.  E.  D. 

In  a  balanced  polyphase  system,  a  condition  which  is  seldom 
strictly  realized,  the  power  taken  by  one  only  of  the  receiving 
circuits  need  be  measured. 


PROBLEMS. 

69.  A  common  return  wire  is  used  for  the  two  currents  of  a 
two-phase  system.     The  system  is  balanced  and  each  current  is 
equal  to  100  amperes.     What  is.  the  current  in  the  common  re- 
turn wire?     Ans.  141.4  amperes. 

70.  The  electromotive  force  of  each  phase,  problem  i,  is  500 
volts.     What   is    the   electromotive  force   between   the   outside 
wires  ?     Ans.  707  volts. 

71.  Three  similar  receiving  circuits  are  A-connected  to  3 -phase 
mains,  the  electromotive  force  between  each  pair  of  mains  being 
no  volts.     The  power  delivered   to  the  three   circuits  is    150 


SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS.      I  13 

kilowatts  and  the  power  factor  of  each  circuit  is  .90.  What  is 
the  current  in  each  circuit  and  in  each  main  ?  Ans.  505  am- 
peres, 885  amperes. 

72.  Three  similar  receiving  circuits  are  Y-connected  to  the  3- 
phase  mains,  problem  71  ;  the  total  power  delivered  is  150  kilo- 
watts and   the  power  factor  of  each  circuit  is  .90.     What  is  the 
current  in  each  circuit  and  in  each  main  ;  and  what  is  the  electro- 
motive force  between  the  terminals  of  each  circuit?     Ans.   885 
amperes,  885  amperes,  63.5  volts. 

73.  A  three-phase  alternator  is  provided  with  4  collecting  rings 
and  4  mains  as  shown  in  Fig.  84.     Three  similar  receiving  cir- 
cuits are  connected  as  follows :  one  from  main  I  to  main  4,  one 
from  main  2  to  main  4,  and  one  from  main  3  to  main  4.     Each 
receiving  circuit  takes  150  amperes.     \Vhen  the  armature  wind- 
ings are  properly  connected  to  the  collector  rings  the  current  in 
main  4  is  zero.     What  is  the  current  in  main  4  when  one  arma- 
ture  winding  has  its  connections   reversed  ?     Draw   a   diagram 
showing  the  phase  relations  between  the  currents  in  mains  I,  2, 
3,  and  4  when  the  armature  winding  A,  Fig.  84,  is  reversed.      In 
constructing  this  diagram  consider  directions  out  from  machine 
as  the  positive  direction  in  each  main.     Ans.   300  amperes. 

74.  The  three  windings  of  a  three-phase  alternator  are  Y-con- 
nected to  three  mains  as  shown  in  Fig.  85.     The  electromotive 
force  of  each  winding  is  1 2  5  volts.     The  connections  of  winding 
A   are    reversed.     Draw  a  diagram   representing    electromotive 
forces  from  main   I  to  main  2,  from  main  2  to  main  3,  and  from 
main  3  to  main  i. 

75.  The  distance  from  center  to  center  of  adjacent  poles  of  an 
alternator  measured  on  the  surface  of  the  armature  is  10.4  inches. 
How  far  apart  must  two  armature  conductors  be  placed  so  that 
there  may  be  a  phase  difference  of  5  5  °  in  the  electromotive  forces 
induced  in  the  respective  conductors?     Ans.   3.18  inches. 


CHAPTER    IX. 

ALTERNATORS. 

( Continued. ) 

79.  Armature  reaction. — The  amount  of  magnetic  flux  entering 
the  armature  core  from  the  field  poles  and  the  manner  of  its  dis- 
tribution over  the  pole  faces,  depend  upon  the  combined  magnetic 
action  of  the  field  current  and  of  the  armature  current. 

Distortion  of  field. — The  armature  current  in  an  alternator  tends 
to  concentrate  the  magnetic  flux  under  the  trailing  horns  of  the 
pole  pieces  very  much  as  in  the  direct-current  dynamo.  The 
effect  of  this  concentration  of  flux  is  to  slightly  increase  the  mag- 
netic reluctance  of  the  saturated  portions  of  the  pole  pieces  and 
armature  core.  This  increase  of  magnetic  reluctance  causes  a 
decrease  of  flux  and  a  consequent  decrease  of  the  electromotive 
force  of  the  alternator,  other  things  being  equal.  This  effect 
may  be  appreciable,  but  it  cannot  be  predetermined  accurately 
by  calculation. 

Magnetizing  and  demagnetizing  action  of  the  armature  current. 
— When  the  current  given  by  an  alternator  is  in  phase  with  its 
electromotive  force  the  only  effect  of  the  armature  current  upon 
the  field  is  the  distorting  effect  described  above.  When  the  cur- 
rent is  not  in  phase  with  the  electromotive  force  the  distorting 
effect  is  decreased  *  and  in  addition  there  is  a  magnetizing  action 
or  demagnetizing  action  upon  the  field  according  as  the  current 
is  aliead  of  or  behind  the  electromotive  force  in  phase. 

*  The  distorting  effect  is  due  to  the  component  of  the  current  parallel  to  the  electro- 
motive force  and  the  magnetizing  effect  is  due  to  the  component  at  right  angles  to  the 
electromotive  force. 

114    • 


ALTERNATORS. 


Consider  a  bundle  of  ./V  armature  wires,  grouped  in  a  slot,  for  example.     Let 

e  =  J^  sin  w/ 

be  the  alternating  electromotive  force  induced  in  this  bundle  of  conductors.  This 
electromotive  force  is  a  maximum  when  the  slot  is  at  a,  Fig.  92.  It  is  zero  when 
the  slot  is  at  b  and  it  is  a  minimum  (negative  maximum)  at  c.  Therefore  the  value 
of  ut  is  90°  at  a,  180°  at  b  and  270°  at  c.  Let  i  be  a  given  current  flowing  in  the 
bundle  of  wires.  The  ampere-turns  of  the  bundle  is  then  Ni.  If  the  bundle  of 
wires  is  at  a  its  ampere  turns  will  be  without  appreciable  effect  on  the  magnetic  cir- 
cuit m  m  m  shown  by  the  dotted  line;  at  b  the  ampere- turns  will  have  their  full 
demagnetizing  effect  (negative)  *  upon  the 
magnetic  circuit,  and  at  c  the  effect  will  again 
be  zero. 

Now,  cos  ut  is  zero  at  a,  negative  unity  at  b 
and  zero  at  c.  Therefore  Ni  cos  «/  is  an  ex- 
pression which  gives  the  true  magnetic  effect  of 
the  bundle  of  wires  at  a,  b  and  c  with  the  given 
current.  We  assume  this  expression  to  hold 
for  all  points  between  a  and  c.  The  actual 
current  in  the  bundle  of  wires  is 

i=  Jsin  (ut— 6) 

in  which  6  is  the  angle  of  lag  of  the  current 
behind  the  electromotive  force.  Therefore 
substituting  this  value  of  i  in  the  expression 
Ni  cos  ut  we  have 

m  =  A7!  cos  w(  sin  (ut  —  6] 

in  which  m  is  the  effective  ampere-turns  of  the 
bundle  of  wires  at  the  instant  /.  To  find  the 
average  value  of  m  expand  sin  («/  —  6), 
whence 

m  =  NI  sin  wt  cos  ut  cos  8  —  NI  cos2  ut  sin  6 


Fig.  92. 


Now  the  average  value  of  m  is  the  sum  of  the  average  values  of  the  two  terms  of 
the  right-hand  member ;  but  the  average  value  of  sin  ut  cos  <•>/  is  zero  and  the  average 
value  of  cos2  ut  is  y^.  Therefore 

average  ;;/  =  —  y2  NI  sin  8  ( ii  ) 

or  putting  V 2  /for  I  we  have 

average  magnetizing  ampere-turns  =  —  0.707  A7"fcin  8  (58) 

When  this  formula  is  applied  to  an  alternator  the  windings  of  which  are  not  too 
widely  distributed,  N  is  to  be  taken  as  the  total  number  of  armature  conductors  divided 
by  the  number  of  poles.  This  equation  shows  that  when  the  current  lags  behind  the 

*  The  current  i  is  considered  positive  when  it  is  in  the  direction  of  the  electromo- 
tive force  which  is  induced  under  the  N  pole.  A  current  in  this  direction  has  a 
demagnetizing  effect  for  all  positions  of  the  slot  between  a  and  c. 


Il6  ELEMENTS   OF   ALTERNATING   CURRENTS. 

electromotive  force  (angle  6  positive)  the  armature  current  weakens  the  field  and  vice 
ver.a. 

Remark. — The  distorting  action  and  the  magnetizing  or  de- 
magnetizing action  of  the  armature  currents  of  a  polyphase 
alternator  supplying  currents  to  a  balanced  receiving  system  are 
steady  in  value,  that  is,  they  do  not  pulsate  as  in  the  single-phase 
alternator. 

In  the  case  of  the  two-phase  alternator  the  constancy  of  the  magnetizing  action 
of  the  armature  currents  is  easily  shown.  Let  m,  equation  (i),  be  the  instantaneous 
magnetizing  action  of  the  armature  currents  in  the  A  winding  of  a  two-phase  machine. 
Since  the  A  and  B  windings  supply  current  to  entirely  similar  receiving  circuits,  I 
and  6  have  the  same  values  for  both  phases,  and  the  instantaneous  value  of  the 
magnetizing  action  of  the  armature  currents  in  the  B  winding  is  obtained  by  substi- 
tuting w/rfc  90°  for  ut  in  equation  (i).  This  gives  : 

in*  •==.  —  N1  sin  uf  cos  ut  cos  6  —  JVI  sin2  of  sin  0  (nO 

The  combined  magnetizing  action  of  the  currents  in  A  and  B  is 

m  -\-m'  =  —  AT  (sin2  ut  -\-  cos2  ut]  sin  6 
or 

m-\-m'  =  —  AT  sin  0  (iv) 

Remark. — The  pulsating  magnetizing  action  of  the  armature 
currents  of  an  alternator  (together  with  the  varying  reluctance 
of  the  magnetic  circuit  in  the  case  of  armatures  with  large  teeth) 
causes  the  field  flux  to  pulsate,  and  this  pulsation  of  field  flux 
induces  double  frequency  alternating  electromotive  forces  in  the 
field  windings.  These  alternating  electromotive  forces  cause  in 
their  turn  a  pulsation  of  the  field  current.  A  direct  current 
ammeter  connected  in  the  field  circuit  indicates  the  average  cur- 
rent, which  is  slightly  less  than  the  square-root-of-mean-squarc 
which  is  indicated  by  an  alternating  current  ammeter.  A  direct- 
current  voltmeter  connected  to  the  terminals  of  a  field  coil  indi- 
cates the  average  value  of  the  electromotive  force,  and  this  is 
sometimes  much  less  than  the  square-root-of -mean-square  value  as 
indicated  by  an  alternating-current  voltmeter. 

80.  Armature  inductance. — The  value  of  the  inductance  of  an 
alternator  armature  varies  with  the  position  of  the  armature  coils 
with  respect  to  the  field  magnet  poles,  so  that  the  inductance  of 


ALTERNATORS.  I  I/ 

an  armature  pulsates  at  a  frequency  twice  *  as  great  as  the 
frequency  of  the  electromotive  force  of  the  alternator.  The 
armature  of  the  alternator  shown  in  Fig.  10,  for  example,  has 
about  two  times  as  great  inductance  when  the  armature  teeth  are 
squarely  under  the  field  poles  as  it  has  when  the  armature  teeth 
are  midway  between  field  poles.  That  is,  the  flux  produced 
through  the  armature  teeth  by  a  given  current  is  three  or  four 
times  as  great  in  the  first  case  as  in  the  second  case.  This  fluc- 
tuation of  armature  inductance  makes  it  very  difficult  to  carry  out 
accurate  calculations  upon  the  action  of  the  machine.  In  the 
following  discussion  the  armature  inductance  is  assumed  to  be 
constant. 

The  inductance  of  an  alternator  armature  is  proportional  to  the 
linear  dimensions  of  the  armature,  other  things  being  equal ;  and 
the  inductance  of  an  armature  of  given  size  is  much  greater  when 
the  winding  is  concentrated  than  it  is  when  the  winding  is  dis- 
tributed. 

Armature  inductance  is  advantageous  in  an  alternator  which  is 
especially  liable  to  be  short  circuited.  The  armature  inductance 
keeps  the  current  from  becoming  excessive.  Armature  induc- 
tance is  more  or  less  objectionable  in  an  alternator  which  is  to  be 
used  to  supply  current  at  constant  electromotive  force  on  account 
of  the  electromotive  force  lost  in  the  armature  as  explained  in 
the  next  article. 

The  inductance  of  an  armature  is  best  determined  by  sending 
a  measured  alternating  current  /  through  it  at  standstill  from  an 
outside  source,  and  measuring  the  electromotive  force  E  between 
the  collector  rings.  Then 


from  which  L  may  be  calculated  when  the  armature  resistance  R 
and  the  frequency  (G>  27r)  are  known.     The  value  of  L  depends 

*The  electromotive  force  of  an  alternator  passes  through  a  cycle  as  an  armature 
coil  passes  from  a  north  pole  of  the  field  to  the  next  north  pole.  The  inductance 
passes  through  a  cycle  of  values  as  an  armature  coil  passes  from  one  field  pole  to  the 
next  field  pole. 


n8 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


greatly  upon  the  position  in  which  the  armature  is  held,  as  ex- 
plained above. 

For  further  information  concerning  armature  inductance,  see 
Parshall  and  Hobart,  "Electric  Generators,"  pages  160  to  175, 
London,  1900,  and  London  Engineering,  Vol.  70,  pages  141  to 
145,  August  3,  1900. 

81.  The  electromotive  force  lost  in  the  armature,  Armature 
drop. — -The  electromotive  force  at  the  collecting  rings  of  an  alter- 
nator is  less  than  the  total  electromotive  force  induced  in  the 
armature,  for  the  reason  that  a  portion  of  the  induced  electromo- 
tive force  is  used  to  overcome  the  resistance,  and  a  portion  is 
also  used  to  overcome  the  inductance  of  the  armature  windings. 
The  numerical  difference  between  the  electromotive  force  induced 

in  the  armature  and  the  elec- 
tromotive force  between  the 
-U)i I  brushes  is  called  the  armature 
drop. 

General  case. — Let  /,  Fig. 
93,  be  the  current  given  by 
the  alternator  and  E  the  total 
induced  electromotive  force. 
Then  o>Z7  is  the  portion  of  E  used  to  overcome  the  inductance  L 
of  the  armature,  and  RI  is  the  portion  of  E  used  to  overcome  the 
resistance  R  of  the  armature.  Subtracting  wLI  and  RI  from 
E  gives  the  electromotive  force  at  the  collecting  rings,  or  "  exter- 
nal electromotive  force,"  Ex. 

Armature  drop,  non-inductive  load. — In  this  case  E  is  nearly  in 
phase  with  /,  and  the  subtraction  of  (oLI  from  E  scarcely  re- 
duces its  value,  &LI  being  nearly  at  right  angles  to  E.  There- 
fore, with  a  non-inductive  receiving  circuit,  the  armature  drop 
depends  almost  wholly  upon  the  armature  resistance. 

Armature  drop,  inductive  load. — When  the  phase  difference  be- 
tween E  and  /is  nearly  90°,  then  the  subtraction  of  RI  from  E 
scarcely  reduces  its  value.  Therefore,  with  a  highly  inductive 


RI 


I 

Fig.  93. 


ALTERNATORS.  119 

receiving  circuit,  the  armature  drop  depends  almost  wholly  upon 
the  armature  inductance. 

Algebraic  expression  for  armature  drop. — Let  /,  Fig.  94,  repre- 
sent the  current  delivered  by  the  alternator.  Let  E  represent 
the  total  electromotive  force  induced  in  the  armature,  let  Ea  be 
the  electromotive  force  lost  in  the  armature,  and  let  Ex  be  the 
electromotive  force  between  the  collecting  rings.  The  electro- 
motive force  E  is  the  vector  sum  of  Ea  and  Ex.  The  angle  9  is 
the  phase  difference  between  Ex  and  /,  and  cos  9  is  the  power 
factor  of  the  receiving  circuit.  The  angle  9f  is  the  phase  differ- 
ence between  E  and  /.  The  electromotive  forces  E  and  E  are 


usually  large  in  comparison  with  £a,  E  and  Ex  are  therefore  ap- 
proximately parallel  to  each  other,  and  the  angle  0  is  approxi- 
mately equal  to  6'  —  6.  The  numerical  difference  between  E 
and  Ex,  the  armature  drop,  is  approximately  equal  to 


E  cos 


-f  o)2Z2-  cos  0  =  IVR2  +  «2Z2  cos  (0'  -  6). 


We  shall  have  occasion  in  the  discussion  of  the  compensated  al- 
ternator of  Mr.  E.  W.  Rice  to  remember  that  the  armature  drop 
is  proportional  to  7  cos  (6'  —  0),  the  factor  \/R2  +  o>2Z2  being  a 
constant  for  a  given  alternator. 

Remark.  —  If  one  considers  the  actual  flux  through  a  circuit, 
due  to  the  combined  action  of  all  causes  which  tend  to  produce 


120 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


flux,  then  the  rate  of  change  of  this  flux  is  an  electromotive 
force,  no  portion  of  which  can  be  lost  in  overcoming  inductance, 
inasmuch  as  the  inductance  flux  will  have  already  been  consid- 
ered. Thus  the  electromotive  force  lost  in  an  armature  because 
of  inductance  may  be  allowed  for,  as  above,  by  subtracting  coLf, 
as  a  vector,  of  course,  from  the  electromotive  force  E,  which 
would  be  produced  by  the  given  field  excitation  with  no  armature 
current,  or  the  demagnetizing  ampere-turns  on  the  armature  may 
be  considered  with  the  ampere-turns  on  the  field  spools,  the  ac- 
tual flux  due  to  both  found,  and  the  electromotive  force  induced 
in  the  armature  by  this  net  flux  will  all  be  available  as  external 
electromotive  force,  except  that  a  portion  of  it  is  lost  in  over- 
coming the  resistance  of  the  armature. 

82.  The  characteristic  curve  of  the  alternator. — The  curve  ob- 
tained by  plotting  observed  values  of  the  external  electromotive 

force  for  various  currents 
taken  from  an  alternator  is 
called  the  characteristic  curve 
of  the  alternator.  Such  char- 
acteristic curves  are  shown  in 
Fig-  95-  Curve  A  is  for  a 
separately  excited  alternator 
having  but  small  armature 
inductance,  and  curve  B  is 
for  a,  separately  excited  alter- 
nator having  large  armature 
inductance.  The  shape  of  the 
characteristic  curve  of  a  given 
alternator  depends  to  a  greater 
or  less  extent  upon  the  inductance  of  the  receiving  circuit.  The 
falling  off  of  electromotive  force  with  increase  of  current  is  due 
in  part  to  the  demagnetizing  action  of  the  armature  current, 
which  weakens  the  field,  and  in  part  to  the  increased  armature 
drop  with  increase  of  current. 


2400 

2200 


1800 


200 
O 


^^ 

--^, 

- 

- 

—     », 

•~~—^, 

3[ 

^^ 

X, 

^> 

x 

\ 

B 

\ 

s 

\ 

\ 

current 

Fig.  95. 


ALTERNATORS. 


121 


Fig.  96. 


83.  The  constant  current  alternator. — An  alternator  of  which 
the  armature  has  an  excessive  inductance,  or  an  ordinary  alterna- 
tor in  circuit  with  which  a  large  inductance  is  connected,  gives  a 
current  which  does  not  vary  greatly  with  the  resistance  *  of  the 
receiving  circuit. 

This  may  be  shown  as  follows  :  Let  E,  Fig. 
96,  be  the  total  induced  electromotive  force  of 
an  alternator  sending  current  through  a  circuit 
of  which  the  reactance  coL  is  constant  and  large, 
compared  with  the  resistance  R.  Then  <oLI 
will  be  large,  compared  with  RI.  Further,  RI 
and  o)LI  are  at  right  angles  to  each  other  and 
their  vector  sum  is  E,  so  that  the  point  P,  Fig. 
96,  lies  on  a  semicircle  constructed  on  E  as  a 
diameter.  Now,  when  RI  is  small,  compared 
with  E,  then  coLI  is  very  nearly  equal  to  E,  that  is,  wLI  is  ap- 
proximately constant  and,  therefore,  /  is  approximately  constant. 

84.  Effect  of  distributed  winding  upon  the  electromotive  force 
of  an   alternator.f — Consider  an   armature  winding  A   concen- 
trated in  a  set  of  slots,  one  slot  per  pole.     The  effective  electro- 
motive force  of  this  winding  is 

4.44$  Tf 
io8 

according  to  equation  (21),  Chapter  II.  Suppose  another  simi- 
lar concentrated  winding,  Bt  is  placed  upon  the  same  arma- 
ture in  slots  distant  s  from  the  first  set  of  slots,  s  being  the  angle 
shown  in  Fig.  97.  This  figure  shows  one  slot  only  of  the  first 
set  and  one  slot  only  of  the  second  set.  The  phase  difference 

*  Unless  the  resistance  becomes  very  large. 

f  This  question  is  discussed  in  a  slightly  different  manner  in  the  chapter  on  the 
rotary  converter.  The  discussion  in  this  article  is  based  upon  the  assumption  that  the 
magnetic  flux  passing  into  (and  out  of)  the  armature  from  the  field  magnet  is  so  dis- 
tributed that  a  harmonic  electromotive  force  is  induced  in  each  armature  conductor. 
The  discussion  for  any  other  type  of  flux  distribution  would  lead  into  the  discussion  of 
non-harmonic  electromotive  forces,  which  is  beyond  the  scope  of  this  text. 


122 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


between  the  electromotive  forces  in  these  two  windings  is  the 

angle-  X  360°,  inas- 
much as  the  angle  q 
from  TV  to  TV7"  is  equiva- 
lent to  360°  of  phase 
difference.  These  two 
electromotive  forces  are 
represented  by  the  lines 
A  and  B,  Fig.  98.  Simi- 
larly the  lines  C  and  D 
represent  the  electro- 
motive forces  in  two 
additional  similar  wind- 
ings concentrated  in 
two  additional  sets  of 
slots  c  and  </,  Fig.  97. 
If  all  these  windings 
are  connected  in  series 
the  effective  electromo- 
tive force  produced  will 
be  the  vector  sum  E  of 
A,  B,  C  and  D.  If  we 

Fig.  97. 

were   to    calculate    the 

effective  electromotive  force  produced  by  A,  B,  C  and  D  in  series 
on  the  assumption  that  all  the  windings  are  concentrated  in  one 


Fig.  98. 


ALTERNATORS. 


123 


set  of  slots,  that  is,  if  we  were  to  calculate  the  total  electromo- 
tive force  by  means  of 
equation  (2 1 ),  using  for 
T  the  total  number  of 
turns  in  all  the  wind- 
ings, we  would  get  a 
result  greater  than  E  in 
the  ratio  of  the  sum  of 
the  sides  A,  By  C  and 
D  of  the  polygon  to 
the  chord  E,  Fig.  98. 
This  ratio  may  be 
called  the  phase  con- 
stant of  the  distributed 
winding.  By  introduc- 
ing the  phase  constant  Fig-  "• 
k  in  equation  (21)  this  equation  becomes 


(59) 


I08 


This  form  of  the  fundamental  equation  of  the  alternator  is  ap- 
plicable to  armatures  with  distributed  windings.  The  following 
table  gives  the  values  of  k  for  various  degrees  of  distribution. 
The  slots  for  a  given  winding  are  always  grouped  so  many  per 
pole  and  a  group  of  slots  may  cover  *^,  ^,  *4,  etc.,  of  the  space 

VALUES  OF  PHASE  CONSTANT  k  FOR  DISTRIBUTED  WINDINGS. 

Widths  of  groups  of  slots  in  fractional  parts  of  N  to  S. 


Number  of  slots 
in  each  group. 

X 

K 

T/ 

K 

Whole. 

2 
3 
4 
Infinity. 

I.OOO 

.980 

.977 
.976 

•975 

I.OOO 

.966 

.960 
.958 

•955 

I.OOO 

.924 
.912 
.908 
.901 

I.OOO 

.831 
.805 

I.OOO 

.707 

.666 
•653 
.637 

Note. — Column  headed  */$  applies  to  3-phase  alternators.  Column  headed  ^  ap- 
plies to  2-phase  alternators.  Width  of  group  =  ns,  where  n  is  number  of  slots  in  a 
group  and  j  is  distance  from  center  to  center  of  adjacent  slots. 


124 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


from  the  center  of  an  N  pole  to  the  center  of  an  5  pole.  Thus 
in  Fig.  99  is  shown  an  8 -pole  machine,  of  which  the  armature  is 
slotted  for  a  distributed  winding,  there  being  three  slots  per  pole, 
these  slots  being  grouped  so  as  to  cover  ^  of  the  space  from  an 
N  to  an  5  pole.  In  the  table  the. width  of  a  group  of  slots  is 
ns  where  n  is  the  number  of  slots  in  a  group  and  s  is  the  distance 
from  center  to  center  of  adjacent  slots. 

85.  Practical  and  ultimate  limits  of  output. — The  dotted  curve, 
Fig.  100,  is  the  characteristic  curve  of  a  given  alternator.  This 

curve  shows  the  relation  be- 
tween the  current  output  and 
the  electromotive  force  be- 
tween the  collecting  rings,  the 
field  excitation  being  kept 
constant.  The  ordinates  of 
the  full-line  curve  represent 
the  power  outputs  corres- 
ponding to  the  different  cur- 
rents  (receiving  circuit  non- 
inductive).  The  maximum 
output  of  the  alternator  is 
thus  68  kilowatts  when  the 
current  output  is  38  amperes. 
Fig'100-  In  practice  the  allowable 

power  output  of  an  alternator  is  limited  to  a  smaller  value  than 
•this  maximum  output  by  one  or  the  other  of  the  following  con- 
siderations : 

(a)  Electric  lighting  and  power  service  usually  demands  an  ap- 
proximately constant  electromotive  force  and  it  is  not  permissible 
to  take  from  an  alternator  so  large  a  current  as  to  greatly  reduce 
its  electromotive  force.  This  difficulty  may  be  largely  overcome 
by  providing  for  an  increase  of  field  excitation  of  the  alternator 
with  increase  of  load  as  is  done  in  the  alternator  with  a  compound 
field  winding.  (See  Article  94.) 


ZfcUU 

£400 
2200 

--- 

-^ 

"^-v 

^x 

Xv 

x 

tflon 

\ 

1600 
1400 

^~, 

\ 

^1200 

^ 

•• 

\\ 

/ 

/ 

1 

800 
600 

/ 

/ 

40o 

/ 

200 

O 

/ 

5     10     15     20    25  30    35    40   45    50 

current 


ALTERNATORS.  125 

(b)  The  current  delivered  by  an  alternator  generates  heat  *  in 
the  armature  of  the  alternator  and  the  temperature  of  the  armature 
rises  until  it  radiates  heat  as  fast  as  heat  is  generated  in  it  by  the 
current.  Excessive  heating  of  the  armature  endangers  the  insu- 
lation of  the  windings  and  it  is  not  permissible  to  take  from  an 
alternator  so  large  a  current  as  to  heat  its  armature  more  than 
40°  or  50°  C.  above  the  temperature,  of  the  surrounding  air. 
This  heating  effect  of  the  armature  currents  usually  fixes  the 
allowable  output  of  an  alternator,  except  in  those  rare  cases  where 
extreme  steadiness  of  electromotive  force  is  required,  or  where 
the  alternator  is  not  compounded. 

Influence  of  inductance  upon  output. — An  alternator  is  rated 
according  to  the  power  it  can  deliver  steadily  to  a  non-inductive 
receiving  circuit  without  overheating.  The  amount  of  power 
which  an  alternator  can  satisfactorily  deliver  to  an  inductive  re- 
ceiving circuit  is  less  than  that  which  it  can  deliver  to  a  non-in- 
ductive receiving  circuit,  because  of  the  phase  difference  of  elec- 
tromotive force  and  current.  The  cosine  of  the  angle  of  phase 
difference  (cos  6)  is  called  the  power  factor  of  the  receiving  circuit 
as  before  pointed  out.  The  power  factor  of  lighting  circuits  is 
very  nearly  unity. 

The  power  factor  of  induction  motors,  synchronous  motors  and 
rotary  converters  is  often  as  low  as  .75  and  sometimes  even  less. 

86,  Frequencies. — The  frequencies  employed  in  practice  range 
from  20  or  25  to  150  cycles  per  second.  Very  low  frequencies 
are  not  suitable  for  lighting  on  account  of  the  tendency  to  pro- 
duce flickering  of  the  lights  ;  on  the  other  hand  high  frequencies, 
which  tend  to  make  transformers  cheaper  for  a  given  output,  are 
entirety  satisfactory  and  are  often  employed  for  lighting. 

High  frequencies  are  not  well  adapted  for  the  operation  of  in- 
duction motors,  synchronous  motors  and  rotary  converters  be- 
cause high  frequencies  necessitate  either  great  speed  or  a  great 

*  Additional  heat  is  generated  in  the  armature  by  the  hysteresis  and  eddy  current 
losses  in  the  armature  core. 


26 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


number  of  poles.     For  such  purposes  frequencies  as  low  as  25 
per  second  are  often  employed. 

A  frequency  of  60  has  been  quite  generally  adopted  for 
machines  used  to  operate  both  lights  and  motors. 

87.  Speeds.  Number  of  poles. — A  machine  which  is  to  be 
belt-driven  may  be  driven  as  fast  as  is  compatible  with  the 
strength  and  rigidity  of  the  rotating  part.  The  allowable  speed 
of  rotation  in  ordinary  dynamos  and  alternators  is  such  as  will 
give  a  peripheral  velocity  of  from  4,000  to  6,000  feet  per  min- 
ute. When  a  machine  is  direct-connected  to  an  engine  or  water- 
wheel  its  speed  is  fixed  by  that  of  the  prime  mover. 

The  number  of  poles  depends  upon  the  speed  of  an  alternator 
and  the  frequency  it  is  to  give,  according  to  equation  ( 1 7).  Large 
machines  as  a  rule  must  run  slower  than  small  ones,  and  they, 
therefore,  have  a  greater  number  of  poles.  The  accompanying 
table  gives  data  as  to  speed,  frequency,  and  number  of  poles  of 
a  few  recent  American  machines.  Machines  7  and  8  are  of  the 
direct-connected  type. 

TABLE. 


125  CYCLES. 

60  CYCLES. 

No. 

No.  of 
poles. 

Output 
K.  W. 

Speed 
r.  p.  m. 

No 

No.  of 
poles. 

Output 
K.  W. 

Speed 
r.  p.  m. 

I 

2 

3 

10 
*4 

16 

60 

125 
200 

1500 
IO7O 

937 

4 
6 

8 

12 

16 
36 
40 

75 
150 
250 
250 

75° 

900 
6OO 

450 
2OO 
1  80 

88.  Armatures. — Alternator  armatures  are  usually  of  the  drum 
type  or  disc  type.  The  former  type  is  almost  universal  in  Amer- 
ica, while  the  disc  type  is  frequently  used  in  England;  for 'exam- 
ple, the  Ferranti  and  Mordey  machines  have  disc  armatures. 
Drum  armatures  have  laminated  iron  cores  similar  to  the  arma- 
ture cores  used  for  direct-current  dynamos,  while  disc  armatures 
are  usually  made  up  without  iron.  Ring  armatures  have  been 
used  only  to  a  very  limited  extent. 


ALTERNATORS.  I2/ 

Drum  armatures  are,  in  nearly  all  modern  machines,  of  the 
toothed  or  ironclad  type.  The  conductors  are  bedded  in  slots. 
This  has  the  double  advantage  of  shortening  the  gap  space  from 
pole  face  to  armature  core  and  of  protecting  the  armature  con- 
ductors from  injury.  One  type  of  such  an  armature  has  already 
been  shown  in  Fig.  10,  the  heavy  coils  being  first  wound  on  forms 
and  then  pressed  into  position  on  the  armature  core.  When  dis- 
tributed windings  are  used  straight  slots  as  shown  in  Fig.  101  are 


Fig.  101.  Fig.  102. 

often  employed.  Fig.  102  shows  a  style  of  slot  commonly  used 
in  which  the  coils  are  held  in  position  by  the  wooden  wedge  IV. 
Armature  core  discs  should  be  varnished,  japanned  or  in  some 
manner  insulated  from  each  other  to  prevent  eddy  currents. 
This  is  especially  necessary  in  the  case  of  alternator  armature 
cores  because  the  frequency  is  comparatively  high. 

89.  Armature  windings. — Any  direct-current  dynamo  *  may 
be  converted  into  a  single-phase  or  polyphase  alternator  by  pro- 
viding it  with  collecting  rings  as  explained  in  the  chapter  on  the 
rotary  converter.  Ordinarily,  however,  the  armature  windings 
of  alternators  are  very  different  from  the  armature  windings 
of  direct-current  dynamos.  In  the  type  of  winding  most  fre- 
quently employed  a  number  of  distinct  coils  are  arranged  on  the 
armature  ;  in  these  coils  alternating  electromotive  forces  are  in- 
duced as  they  pass  the  field  magnet  poles,  and  these  coils  are 
connected  in  series  between  the  collecting  rings  if  high  electro- 
motive force  is  desired,  or  in  parallel  f  between  the  collecting 
rings  if  low  electromotive  force  is  desired. 

*  Except  the  so-called  unipolar  dynamo. 

•f  The  coils  of  a  distributed  winding  cannot  all  be  connected  in  parallel  between 


128 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


Single-phase  winding. — Fig.  10  shows  a  common  type  of  sin- 
gle-phase winding  having  one  coil  per  pole.  Fig.  103  shows 
another  type  of  concentrated  single-phase  winding  having  one 
coil  to  each  pair  of  poles  or  one  slot  per  pole.  In  the  diagram, 
Fig.  103,  the  heavy  sector-shaped  figures  represent  the  coils  and 
the  light  lines  represent  the  connections  between  the  terminals  of 
the  coils.  The  radial  parts  of  the  sector-shaped  figures  repre- 


Fig.  103. 

sent  the  portions  of  the  coils  which  lie  in  the  slots,  and  the 
curved  parts  represent  the  ends  of  the  coils.  The  circles  at  the 
center  of  the  figure  represent  the  collecting  rings,  one  being 
shown  inside  the  other  for  clearness.  The  arrows  represent  the 
direction  of  the  current  at  a  given  instant.  All  electromotive 

the  collecting  rings  for  the  reason  that  the  induced  electromotive  forces  in  the  various 
coils  are  not  exactly  in  phase  and  local  currents  would  circulate  in  the  coils  if  con- 
nected in  parallel. 


ALTERNATORS.  1 29 

forces  under  N  poles  are  in  one  direction  and  all  electromotive 
forces  induced  under  5  poles  are  in  the  opposite  direction.  These 
remarks  apply  to  Figs.  103  to  no  inclusive.  Fig.  104  repre- 
sents a  single-phase  winding  distributed  in  tw©  slots  per  pole,  all 
the  coils  being  connected  in  series.  Fig.  104  is  a  type  of  wind- 
ing which,  for  the  same  number  of  conductors,  has  a  smaller 
inductance  than  the  type  shown  in  Fig.  103  and  the  armature 


Fig.  104. 

shown  in  Fig.  104,  for  the  same  number  of  conductors,  gives  a 
smaller  electromotive  force  than  the  armature  shown  in  Fig.  103. 
Two-phase  windings. — The  two-phase  winding  is  two  inde- 
pendent single-phase  windings  on  the  same  armature,  each  being 
connected  to  a  separate  pair  of  collecting  rings,  as  shown  in  Figs. 
105  and  1 06.  Fig.  105  shows  a  two-phase  concentrated  wind- 
ing, one  slot  per  pole  for  each  phase.  Fig.  106  shows  a  two- 
phase  winding  distributed  in  two  slots  per  pole  for  each  phase. 


Fig.  105. 


(130) 


Fig.  106. 


Fig.  108. 


(132) 


Fig.  110. 


ALTERNATORS.  133 

Three-phase  windings. — The  three-phase  winding  is  three  inde-» 
pendent  single-phase  windings  on  the  same  armature,  the  termi- 
nals of  the  individual  windings  being  connected  according  to  the 
Y  scheme  or  A  scheme,  as  explained  in  Article  72.  Fig.  107 
shows  a  three-phase  concentrated  winding,  one  slot  per  pole  for 
each  phase  ;  Y-connected.  Fig.  108  shows  the  same  winding  A- 
connected.  The  Y  connection  gives  v/3  times  as  much  electro- 
motive force  between  collecting  rings  as  the  A  connection  for  the 
same  winding.  The  Y  connection  is  more  suitable  for  high  elec- 
tromotive force  machines  and  A  connection  for  machines  for 
large  current  output.  The  line  current  is  \/3  times  as  great  as 
the  current  in  each  winding  in  a  A-connected  armature.  Fig. 
109  shows  a  three-phase  bar  *  winding  distributed  in  two  slots 
per  pole  for  each  phase.  Fig.  110  shows  a  three-phase  coil 
winding  distributed  in  two  slots  per  pole  for  each  phase  and  ar- 
ranged in  two  layers,  there  being  as  many  coils  on  tne  armature 
as  there  are  slots,  so  that  portions  of  two  coils  lie  in  each  slot, 
one  above  the  other.  The  portions  of  the  coils  represented  by 
full  lines  lie  in  the  upper  parts  of  the  slots  and  the  adjacent 
dotted  portions  lie  in  the  bottoms  of  the  same  slots. 

The  Y-connection. — The  terminals  of  the  individual  windings  which  are  to  be  con- 
nected to  the  common  junction  and  to  the  collecting  rings  may  be  determined  as 
follows  :  Consider  the  instant  when  winding  A  is  squarely  under  the  pole  as  shown 
in  Fig.  107  ;  the  electromotive  force  in  this  winding  (and  current  also  if  the  circuit  is 
non-inductive)  is  a  maximum  and  the  currents  in  the  other  two  phases  B  and  Care 
half  as  great.  If  winding  A  is  connected  so  that  its  current  is  flowing  away  from  K, 
windings  B  and  C  must  be  connected  so  that  their  currents  flow  towards  K. 

The  ^-connection. — The  three  windings  form  a  closed  circuit  when  A-connected. 
The  total  electromotive  force  around  this  circuit  at  any  instant  must  be  zero.  There- 
fore the  electromotive  force  in  winding  A  when  it  is  directly  under  the  poles  must 
oppose  the  electromotive  forces  of  windings  B  and  C. 

90,  Insulation  of  armatures. — Armatures  for  alternators  must 
be  well  insulated  in  cases  where  the  electromotive  force  generated 
is  high,  as  the  electromotive  force  tending  to  break  down  the 

*  One  conductor  in  each  slot.  This  conductor  is  usually  in  the  form  of  a  copper 
bar  of  rectangular  cross  section. 


134  ELEMENTS   OF   ALTERNATING    CURRENTS. 

insulation  is  the  maximum  value  of  the  electromotive  force  gen- 
erated, and  this  is  considerably  greater  than  the  rated  or  effective 
electromotive  force.  Concentrated  or  partially  distributed  wind- 
ings admit  of  quite  a  high  degree  of  insulation,  inasmuch  as  the 
slots  may  be  made  quite  large  and  there  are  comparatively  few 
crossings  of  the  coils  at  the  ends  of  the  armature.  Distributed 
windings  can  not  be  so  highly  insulated  because  there  are  many 
crossings  of  the  coils  and  the  slots  are  necessarily  small  ;  such 
windings  are,  therefore,  not  suitable  for  the  generation  of  high 
electromotive  forces.  Alternators  having  this  type  of  winding 
should  therefore  be  used  in  connection  with  step-up  transformers 
if  a  high  electromotive  force  is  desired.  When  it  is  desired  to 
generate  a  high  pressure  directly,  it  is  best  to  use  a  machine  with 
a  stationary  armature.  Such  armatures  have  been  built  for  elec- 
tromotive forces  of  8,000  or  10,000  volts,  thus  doing  away  with 
the  necessity  of  step-up  transformers  for  power  transmission  lines 
of  moderate  length.  There  is  usually  more  room  for  thorough 
insulation  on  such  armatures  and  the  insulation  is  less  liable  to 
deteriorate  as  it  is  not  disturbed  in  any  way  by  motion  of  the 
armature.  Moreover,  the  use  of  the  stationary  armature  does 
away  with  collector  rings  and  brushes  (for  the  armature)  and  the 
consequent  necessity  of  their  insulation  for  high  potentials. 

The  individual  coils  of  an  alternator  armature  are  generally 
heavily  taped  and  treated  with  insulating  oil  or  varnish,  the  slots 
are  lined  with  heavy  tubes  built  up  of  paper  and  mica  and  all 
parts  of  the  core  which  are  near  the  coils  are  also  covered  with  a 
heavy  layer  of  insulating  material. 

91.  Magnetic  densities  in  armature  and  air  gap, — The  arma- 
ture core  is  usually  made  of  sufficient  cross  section  to  insure  a 
fairly  low  magnetic  density.  This  is  done  in  order  to  keep  down 
the  hysteresis  and  eddy  current  losses  which  would  otherwise  be 
high  on  account  of  the  comparatively  high  frequencies  employed. 
The  allowable  magnetic  density  in  <iie  armature  core  depends 
largely  upon  the  frequency,  since  the  density  for  a  given  loss  may 


ALTERNATORS.  135 

be  higher  the  lower  the  frequency.  The  following  table  from 
Kolben  gives  values  of  the  density  suitable  for  various  frequencies; 

B.    Lines  per  cm2. 

40  cycles 6,500  to  5,500 

50      " 6,000  "   5,000 

60      "     5,000  "  4,500 

80      "     4,5oo  "  4,000 

100      "     4,000  "  3,500 

120      "     i 3,500  "  3,000 

The  allowable  magnetic  density  in  the  air  gap  will  depend  to 
some  extent  upon  the  material  used  for  the  pole  pieces.  With 
cast-iron  poles  this  density  should  not  exceed  4,000  to  4,500 
lines  per  sq.  cm.;  with  wrought-iron  pole  pieces  it  may  be  as 
high  as  6,000  to  7,000  lines  per  sq.  cm. 

92.  Current  densities. — The  current  density  in  early  alternator 
armatures  was  often  very  high,  not  more  than  300  circular  mils 
per  ampere  being  allowed  in  many  cases.     Such  armatures  usually 
ran  very  hot  at  full  load.     The  current  densities  used  in  modem 
machines  are  much  lower,  from  500  to  700  circular  mils  per  am- 
pere being  allowed,  as  in  the  case  of  direct  current  machines.    The 
armature  conductor  is  usually  of  ordinary  cotton-covered  mag- 
net wire  in  the  smaller  machines,  and  when  a  conductor  of  con- 
siderable cross  section  is  required  a  number  of  wires  are  grouped 
in  multiple.      In  larger  machines  copper  bars  are  frequently  used, 
as  these  admit  of  a  large  cross  section  being  put  in  a  minimum 
space.     Wire  of  rectangular  cross  section  and  copper  ribbon  are 
also  used  in  some  cases. 

93,  Outline  of  alternator  design. — An  alternator  is  usually  de- 
signed to  give  an  electromotive  force  of  prescribed  value  and 
frequency,  arid  to  be  capable  of  delivering  a  prescribed  current 
without  undue  heating.     To  design  an  alternator  is  to  so  pro- 
portion the  parts  as*  to  satisfy  the  following  conditions  : 

(a)  The  product  of  revolutions  per  second  into  the  number  of 


136  ELEMENTS   OF   ALTERNATING   CURRENTS. 

pairs  of  field  magnet  poles  must  equal  the  prescribed  frequency 
according  to  equation  (17). 
(b)  Equation  (59)  namely 

77      4-44 


io8 

must  be  satisfied,  to  give  the  prescribed  electromotive  force. 

(c)  Peripheral  speed  of  armature  must  not  exceed  allowable 
limits. 

(d)  The  armature  must  have  sufficient  surface  to  radiate  the 
total  watts  lost  in  the  armature  (including  eddy  current  and  hys- 
teresis losses)  without  excessive  rise  of  temperature. 

There  are  two  distinct  cases  in  the  designing  of  an  alternator, 
as  follows  : 

Case  I. — Where  the  speed  is  fixed  by  independent  considera- 
tions, as,  for  example,  in  direct-connected  machines.  In  this 
case  the  number  of  poles  is  determined  by  the  given  speed  and 
prescribed  frequency.  The  diameter  of  the  armature  follows 
from  the  allowable  peripheral  speed.*  Assuming  from  2%  to 
5  %  f  of  total  rated  output  as  armature  loss,  the  approximate 
length  of  the  armature  is  then  determined  by  the  radiating  sur- 
face required.  A  well -ventilated  armature  should  radiate  from 
.05  to  .06  watt  per  square  inch  (of  cylindrical  surface)  per  de- 
gree Centigrade  rise  of  temperature.  This  constant  of  radiation 
varies  greatly  with  the  style  of  construction  of  the  armature  and 
with  peripheral  speed.  The  length  may  be  slightly  modified 
when  condition  ($)  comes  to  be  considered.  The  flux  ^E>  is  de- 
termined from  the  flux  density  in  the  air  gap  and  the  area  of  each 
pole  face.  The  combined  area  of  the  pole  faces  is  usually  about 
equal  to  half  the  cylindrical  surface  of  the  armature,  or,  in  other 
words,  the  distance  between  tips  of  adjacent  poles  is  equal  to  the 
breadth  of  the  pole  face.  The  number  of  armature  turns  T  is  then 
determined  from  equation  (59).  The  armature  turns  thus  deter- 

*  Direct-connected  dynamos  are  scarcely  ever  run  up  to  the  allowable  peripheral 
speed.  Speeds  from  2,200  to  2,600  feet  per  minute  are  usual. 

~\  According  to  size  of  machine,  low  percentage  being  for  large  machines. 


ALTERNATORS.  137 

mined  may  come  out  an  odd  or  a  fractional  number  and  must  be 
adjusted  to  suit  the  type  of  winding  employed,  that  is,  to  give  the 
required  number  of  coils  each  having  the  same  number  of  turns. 
The  length  of  the  armature  may  then  be  changed  slightly  to 
adjust  3>  so  that  the  required  electromotive  force  will  be  pro- 
duced with  the  adopted  number  of  armature  turns.  The  area  of 
cross  section  of  the  armature  conductors  is  fixed  by  the  allow- 
able current  density  and  rated  current  output. 

Case  II. — When  speed  is  not  fixed  by  independent  considera- 
tions. In  this  case  a  trial  combination  of  poles  and  speed  is 
adopted,  giving  a  speed  suitable  for  the  size  of  the  armature. 
The  remainder  of  the  design  is  then  worked  out  as  above. 

Remark. — When  a  machine  is  provisionally  designed  the  details 
of  its  behavior  may  be  approximately  calculated  without  difficulty, 
and  refinement  of  design  is  attained  by  working  out  a  number  of 
provisional  designs  and  calculating  the  details  of  their  action  ; 
then  the  most  satisfactory  design  may  be  recognized  and  adopted. 

The  proportioning  of  the  magnetic  circuits  and  the  calculation 
of  field  windings  of  an  alternator  is  carried  out  in  the  same  gen- 
eral way  as  in  the  case  of  a  direct-current  dynamo. 

Remark. — In  designing  a  two-phase  alternator  each  winding  is 
allowed  to  cover  half  of  the  armature  surface.  In  designing  a 
three-phase  alternator  each  winding  is  allowed  to  cover  one-third 
of  the  armature  surface. 

94.  Field  excitation  of  alternators. — The  use  of  an  auxiliary 
direct-current  dynamo  for  exciting  the  field  of  an  alternator  has 
been  pointed  out  in  Article  20.  The  electromotive  force  of  an 
alternator  excited  in  this  way  falls  off  greatly  with  increasing 
current  output,  and  to  counteract  this  tendency  an  auxiliary  field 
excitation  is  frequently  provided  which  increases  with  the  current 
output  of  the  machine.  For  this  purpose  the  whole  or  a  portion 
of  the  current  given  out  by  the  machine  is  rectified  *  and  sent 
through  the  auxiliary  field  coils. 

*  Connections  to  field  coils  are  reversed  with  every  reversal  of  main  current  so 
that,  in  the  field  coils,  the  current  is  unidirectional. 


138 


ELEMENTS  OF  ALTERNATING  CURRENTS. 


Fig.  1 1 1  shows  an  alternator  A  with  its  field  coils  F  separately 
excited  from  a  direct-current  dynamo  E.     The  two  rheostats  R 


(JM^JM 
/ 


Fig.  111. 

and  r,  in  series  with  the  alternator  and  exciter  fields  respectively, 

are  used  to  regulate  the  field 
current.  Fig.  1 1 2  shows  an 
alternator  with  two  sets  of 
field  coils  F  and  C.  The  coils 
F  are  separately  excited  as 
before.  The  coils  C,  known 
as  the  series  or  compound 
coils,  are  excited  by  the 
main  current  from  the  alter- 
nator. One  terminal  of  the 
armature  winding  is  con- 
nected directly  to  a  collect- 
ing ring.  The  other  arma- 
ture terminal  connects  to 
one  set  of  bars  in  the  recti- 
fying commutator  B.  From 
the  rectifier  the  current  is 

led  through  the  winding   C,  thence  back  to  the  rectifier,  and 


ALTERNATORS. 


139 


thence  to  the  second  collecting  ring.  The  rectifying  com- 
mutator B  is  provided  with  as  many  segments  as  there  are 
poles  on  the  machine.  The  commutator  reverses  the  connec- 
tions of  the  terminals  of  the  coils  C  at  every  pulsation  of  the 
alternating  current  so  that  the  current  flows  in  C  always  in 
the  same  direction.  The  commutator  B  is  fixed  to  the  armature 
shaft.  A  shunt  s  moving  with  the  commutator  is  sometimes 
used  when  it  is  desired  to  rectify  only  a  portion  of  the  cur- 
rent. A  stationary  shunt  /  is  also  frequently  used  to  regulate 
the  amount  of  current  flowing  around  the  coils  C,  thus  giving  a 
method  of  adjusting  the  compounding. 

Fig.  1 1 3  shows  an  alternator  A  with  two  sets  of  field  coils  F 
and  C  as  before.  One  armature  terminal  is  connected  to  a  col- 
lecting ring,  and  the 
other  armature  terminal 
connects  to  the  primary 
of  a  transformer  T  and 
thence  to  the  other  col- 
lecting ring.  The  ter- 
minals of  the  secondary 
coil  of  T  connect  to  the 
bars  of  the  rectifying 
commutator  B,  from 
which  the  compound 
field  winding  C  is  sup- 
plied. The  transformer 
T  is  usually  placed  in- 
side the  armature.  All 
three  of  the  methods, 
shown  in  Figs.  1 1 1 ,  112 
and  113,  are  in  common 
use  for  field  excitation 
of  alternators.  Compounding  is  necessary  only  with  alternators 
which  have  fairly  high  armature  inductance,  and  which,  with 
constant  field  excitation,  would  give  poor  regulation.  For  low 


Fig.  113. 


140  ELEMENTS   OF   ALTERNATING   CURRENTS. 

inductance  machines  the  separate  excitation  alone  is  usually 
sufficient. 

The  compensated  alternator  of  the  General  Electric  Company. — 
An  alternator  with  a  compound  field  winding  can  be  adjusted  so 
that  the  armature  drop  is  compensated  by  the  compound  wind- 
ing, provided  the  armature  drop  is  proportional  to  the  current 
and  varies  only  with  the  current.  In  Article  8 1  it  is  shown  that 
the  armature  drop  is  proportional  to  /cos  (Q1 '—  0).  See  discus- 
sion of  algebraic  expression  in  Article  8 1  for  explanation  of  /,  Of 
and  6.  A  variation  of  the  power  factor  of  the  receiving  circuit 
(cos  0)  produces,  therefore,  variations  of  armature  drop  which 
cannot  be  compensated  by  a  compound  field  winding.  However, 
an  auxiliary  field  excitation  proportional  to  /cos  (#'—#)  could 
be  adjusted  so  as  to  completely  compensate  for  armature  drop. 
This  condition  is  realized  in  the  compensated  alternator  of  Mr.  E. 
W.  Rice. 

The  action  of  this  compensated  alternator  depends,  in  the  first 
place,  upon  the  fact  that  a  direct-current  dynamo  (the  exciter)  is 
an  alternator,  as  well  as  a  direct-current  dynamo,  as  explained  in 
Chapter  XIII.;  in  the  second  place,  upon  the  fact  that  the  cur- 
rent or  currents  delivered  by  the  alternator  may  be  passed 
through  the  exciter  armature,  so  as  to  exert  upon  the  exciter 
field  a  magnetizing  action  proportional  to  /  cos  (0'  —  0)  if  the 
exciter  is  driven  at  exactly  the  same  frequency  as  the  main  alter- 
nator and  if  the  phase  relation  between  the  alternating  electro- 
motive force  of  the  exciter  and  of  the  main  alternator  is  properly 
adjusted ;  and  in  the  third  place  upon  the  fact  that  the  variation 
thus  produced  in  the  field  magnetization  of  the  exciter  produces 
a  corresponding  variation  of  the  direct  electromotive  force  of  the 
exciter,  a  corresponding  variation  of  its  direct-current  output  and 
a  corresponding  variation  of  field  excitation  of  the  main  alternator. 

The  commercial  form  of  Rice's  alternator  is  usually  a  polyphase 
alternator  with  the  exciter  armature  rigidly  connected  to  the 
shaft  of  the  main  alternator.  The  following  description  applies 
to  the  case  in  which  the  exciter  armature  is  mounted  upon  the 


ALTERNATORS.  141 

main  shaft  and  rotates  with  the  main  armature.  The  polyphase 
currents  from  the  main  armature  pass  by  direct  connection 
through  the  exciter  armature  and  thence  by  way  of  collecting 
rings  to  the  polyphase  receiving  system. 

Imagine  the  main  armature  to  be  standing  in  the  position  in 
which  the  polyphase  electromotive  forces  of  the  main  generator 
are  at  their  maximum  values.  Then  the  phase  angle  <f>,  that  the 
electromotive  forces  of  the  exciter,  considered  as  an  alternator  (its 
electromotive  forces  being  the  electromotive  forces  between  the 
points  at  which  the  main  alternating  currents  enter  the  exciter 
armature),  lag  behind  the  electromotive  forces  of  the  main  alter- 
nator, depends  upon  the  position  of  the  exciter  field  magnet 
poles ;  and  this  angle  can  be  changed  at  will  by  turning  the 
exciter  field  magnet  about  the  axis  of  the  machine. 

Now  the  polyphase  currents  from  the  main  alternator  lag  0° 
behind  the  external  electromotive  forces  of  the  main  alternator, 
so  that  the  alternating  currents  which  pass  through  the  exciter 
armature  are  (<f>  —  6)°  ahead  of  the  exciter  electromotive  forces 
in  phase.  Therefore  these  currents  have  upon  the  exciter  field  a 
magnetizing  action  proportional  to  7  sin  (<f>  —  6\  according  to 
Article  79.  If  the  exciter  field  be  turned  until  <£  =  90°  —  0', 
then  sin  (</>  —  6}  equals  cos  (6r  —  0)  and  the  above  expression  for 
magnetizing  action  becomes  /  cos  (#'  —0).  But  this  is  propor- 
tional to  the  armature  drop,  according  to  Article  81.  Therefore 
the  magnetizing  action  of  the  main  alternating  currents  upon  the 
exciter  field  is  proportional  to  the  armature  drop  in  the  main 
alternator. 

PROBLEMS. 

76.  A  ten -pole  alternator  having  720  armature  conductors  sup- 
plies 30  amperes  to  a  receiving  circuit  of  which  the  power  factor 
is  0.866.     Calculate  the  average  demagnetizing  ampere  turns  due 
to  the  armature  current.     Ans.  763  ampere  turns. 

77.  An  alternator  with  a  toothed  armature  like  Fig.  10,  has 


142  ELEMENTS   OF   ALTERNATING   CURRENTS. 

its  armature  fixed  with  armature  teeth  squarely  under  field  poles. 
26.7  amperes  of  6o-cycle  current  are  passed  through  the  arma- 
ture from  an  auxiliary  source,  and  the  electromotive  force  be- 
tween the  collector  rings  is  65  volts. 

When  the  armature  is  held  so  that  the  armature  teeth  are  mid- 
way between  the  field  poles,  the  electromotive  force  between  the 
collector  rings  is  43  volts  with  the  same  current  as  before. 

When  34  amperes  of  direct  current  are  sent  through  the  arma- 
ture, the  electromotive  force  between  the  collector  rings  is  ob- 
served to  be  1 6  volts.  What  is  the  inductance  of  the  armature 
in  each  position?  Ans.  0.006337  henry,  0.004083  henry. 

78.  Calculate  for  the  above  alternator  at  a  frequency  of  133 
cycles  per  second,  the  armature  drop  for  various  power  factors 
of  receiving  circuit  and  for  a  fixed  current  of  30  amperes.     As- 
sume the  total  electromotive  force  to  be  so  great  that  it  may  be 
assumed  to  be  in  phase  with  the  external   electromotive  force. 
Plot  a  curve  showing  the  results,  representing  values  of  power 
factor  by  abscissas  and  armature  drops  by  ordinates. 

79.  A  so-called  constant  current  alternator  generates  a  total 
electromotive  force  of  1,100  volts  and  has  an  armature  inductance 
of  o.  12  henry  or,  at  its  rated  frequency,  an  armature  reactance 
of  94  ohms.     Calculate  the  current  delivered  by  the  alternator 
to  a  non-inductive  receiving  circuit  of  which  the  resistance  is  zero, 
10  ohms,  20  ohms,  and  30  ohms  respectively,  neglecting  the  re- 
sistance of  the  armature.     Ans.  11.7,  11.63,   TI-45>  and   11.15 
amperes. 

80.  A  three-phase  alternator  has  its   windings  connected  in 
series  as  a  single-phase  winding.     What  is  its  phase  constant  ? 
Ans.  zero  or      . 


CHAPTER   X. 


THE   TRANSFORMER. 

95.  The  transformer  consists  of  a  laminated  iron  core  upon 
which  two  separate  and  distinct  coils  of  wire  are  wound.  Alter- 
nating current  is  supplied  to 
one  of  these  coils  from  an 
alternator  or  other  source. 
This  alternating  current  pro- 
duces  rapid  reversals  of  mag- 
netization of.  the  iron  ;  and 
these  magnetic  reversals  in- 
duce an  alternating  electro- 
motive force  in  the  other  coil, 
which  delivers  alternating 
current  to  a  receiving  circuit. 
The  coil  to  which  the  alter- 
nating current  is  supplied  is 
called  the  primary  coil,  and 
the  coil  which  delivers  cur- 
rent to  a  receiving  circuit  is 
called  the  secondary  coil. 

Fig.  114  shows  the  essen- 
tial features  of  a  commercial 
type  of  transformer.  The 
iron  core  forms  a  closed  mag- 
netic circuit.  The  primary 

coil  is  in  two  parts,  PP  and  P P' ,  one  of  which,  PP,  is  shown  in 
section.     The  secondary  also  consists  of  two  parts  wound,  in 

143 


144  ELEMENTS   OF   ALTERNATING   CURRENTS. 

this  instance,  underneath  the  primary.  One  part  >S.S  of  the  sec- 
ondary is  shown  in  section. 

Step-up  and  step-down  transformation. — Usually  one  coil  of  a 
transformer  has  many  more  turns  of  wire  than  the  other.  When 
the  coil  of  few  turns  is  the  primary  coil  the  transformer  takes 
large  current  at  low  electromotive  force  and  delivers  small  cur- 
rent at  high  electromotive  force.  This  is  called  step-up  trans- 
formation. 

When  the  coil  of  many  turns  is  the  primary  the  transformer 
takes  small  current  at  high  electromotive  force  and  delivers  large 
current  at  low  electromotive  force.  This  is  called  step-down 
transformation.  The  object  of  step-up  and  step-down  trans- 
formation is  explained  in  Article  2 1 . 

96.  The  action  of  the  transformer. — In  the  following  discussion 
N'  represents  the  number  of  turns  of  wire  in  the  primary  coil 
and  N"  represents  the  number  of  turns  of  wire  in  the  secondary 
coil.  The  effect  of  the  resistance  of  the  coils  is  usually  quite 
small  and  it  is  ignored  in  the  present  discussion. 

Ratio  of  primary  current  to  secondary  current. — Aside  from 
resistance,  the  only  thing  which  opposes  the  flow  of  current 
through  the  primary  coil  is  the  reacting  electromotive  force  in- 
duced in  the  primary  coil  by  the  reversals  of  magnetization  of  the 
core.  The  greater  the  range  of  this  magnetization  the  greater 
the  value  of  the  reacting  electromotive  force.  The  combined  mag- 
netizing action  of  the  primary  and  secondary  coils  is  always  such  as 
to  magnetize  the  core  to  that  degree  which  will  make  the  reacting 
'electromotive  force  in  the  primary  coil  equal  to  the  electromotive 
force  of  the  alternator  which  is  forcing  current  through  the  primary 
coil.  Action  is  equal  to  reaction. 

When  the  secondary  coil  is  on  open  circuit,  just  enough  cur- 
rent flows  through  the  primary  coil  to  produce  the  degree  of 
magnetization  above  specified.  Let  this  value  of  the  primary 
current,  which  is  called  the  magnetizing  current,  be  represented 
by  m.  When  current  I"  is  taken  from  the  secondary  coil,  addi- 


THE   TRANSFORMER.  145 

<; 
tional  current  I1 ',  over  and  above  m,  flows  through  the  primary 

coil.  The  current  m  still  suffices  to  magnetize  the  core,  and  the 
magnetizing  action  of  I"  is  exactly  neutralized  by  the  equal  and 
opposite  magnetizing  action  of  /'.  The  magnetizing  action  of  I" 
is  measured  by  the  product  N" I" ,  and  the  magnetizing  action 
of  /'  is  measured  by  the  product  N1 P ',  so  that,  ignoring  alge- 
braic signs,  we  have 

NT  =  N"I" 
or 

/'       N" 

7"=  A^7 

Ratio  of  primary  electromotive  force  to  secondary  electromotive 
force. — The  rapid  reversals  of  magnetization  of  the  iron  core  in- 
duce a  certain  electromotive  force  a  in  each  turn  of  wire  sur- 
rounding the  core.  Therefore  the  total  electromotive  force 
induced  in  the  primary  coil  is  N'a.  This  is  the  reacting  electro- 
motive force  in  the  primary  coil  and  it  is  equal  and  opposite, 'as 
pointed  out  above,  to  the  electromotive  force  E1  which  is  push- 
ing current  through  the  primary  coil ;  so  that,  ignoring  signs,  we 
have 

E'  =  N'a 

Similarly,  the  total  electromotive  force,  En ',  induced  in  the 
secondary  coil  is 

E"  =  N"a 
Therefore 

E'       N' 
WJ=W' 

Remark. — The  above  discussion  should  in  strictness  refer  pri- 
marily to  instantaneous  values  of  /'  and  I"  and  to  instantaneous 
values  of  E'  and  E".  Thus  i'  and  i"  are  at  each  instant  op- 
posite to  each  other  and  in  the  ratio  N"jNr ;  and  e'  and  e"  are 
at  each  instant  opposite  to  each  other  and  in  the  ratio  N' JN". 
Therefore  I'  and  I"  are  opposite  or  180°  apart  in  phase  and  E' 
and  E"  are  opposite  or  180°  apart  in  phase. 

Approximate  equality  of  input  and  output  of  power. — If  the 


146 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


secondary  coil  of  a  transformer  delivers  current  to  an  inductive 
receiving  circuit,  then  E"  and  I"  will  differ  in  phase  by  a  certain 
angle  6 ;  and  since  E1  and  I'  are  opposite  to  E"  and  I"  respec- 
tively, therefore  the  phase  difference  between  E'  and  I'  is  also  6. 
The  power  output  of  the  transformer  is  E" I"  cos  0,  the  power 
intake  is  E' I'  cos  6,  and  these  are  equal  since  E' I'  =  E" I"  ac- 
cording to  equations  (60)  and  (61).  The  resistance  of  the  coils 
is  here  ignored  and  the  energy  taken  from  the  supply  dynamo 
by  the  magnetizing  current  m  is  not  considered. 

97.  Particular  cases  (for  harmonic  electromotive  force  and  cur- 
rent), i.  Non-inductive  receiving  circuit. — In  this  case  E"  and 
I"  are  in  phase  and  therefore  E'  and  /'  are  in  phase  also.  The 
state  of  affairs  is  represented  in  Fig.  115.  The  line  0<&  repre- 


O 


I1 


Of. 


E" 


Fig.  115. 


Fig.  116. 


sents  the  harmonically  varying  core  flux,  OE'  represents  the 
electromotive  force  acting  on  the  primary  and  OE"  represents  the 
electromotive  force  induced  in  the  secondary. 

2.   Inductive  secondary  receiving  circuit. — In  this  case  I"  lags 

behind  E"  by  the  angle  whose  tangent  is  •  D ,  where  L  is  the  in- 

K 


THE   TRANSFORMER. 


ductance  and  R  the  resistance  of  the  receiving  circuit.  Also  /' 
lags  behind  E'  by  the  same  angle.  The  state  of  affairs  is  shown 
in  Fig.  1 1 6. 

3.  Receiving  circuit  containing  a  condenser. — In  this  case  I"  is 

ahead  of  E"  by  the  angle  whose  tangent  is  (  —~  —  o>L  J  -=-  R, 

where  C  is  the  capacity  of  the  condenser,  L  is  the  inductance  of 
the  connecting  wires  and  R  the  irresistance.  Also,  /'  is  ahead  of 
E'  by  the  same  angle.  The  state  of  affairs  is  shown  in  Fig.  1 1 7. 


Fig    118. 

98.  Equivalent  resistance  and  reactance  of  a  transformer  feeding 
a  given  receiving  circuit. — The  primary  of  a  transformer  takes 
from  the  mains  a  definite  current  at  a  definite  phase  lag  when  the 
secondary  is  supplying  current  to  a  given  circuit.  Consider  a 
simple  circuit  of  resistance  r  and  reactance  x  which,  connected  to 
the  mains,  takes  the  snme  current  as  the  primary  of  the  transformer 
and  at  the  same  phase  lag.  The  circuit  is  said  to  be  equivalent  to 
the  transformer  and  its  secondary  receiving  circuit,  and  r  and  x 
are  called  the  equivalent  primary  resistance  and  reactance  respec- 
tively of  the  secondary  receiving  circuit. 


148  ELEMENTS    OF   ALTERNATING   CURRENTS. 

Resolve  the  primary  electromotive  force  E't  Fig.  1 1 8,  into 
components  parallel  to  /'  and  perpendicular  to  /'  as  shown. 
The  component  parallel  to  /'  is  rl'  and  the  component  perpen- 
dicular to  /'  is  xF .  The  triangle  whose  sides  are  Er ,  rlf  and 
xF  is  similar  to  the  triangle  whose  sides  are  En \  RF'  and  XFf, 
R  and  X  being  the  given  resistance  and  reactance  of  the  second- 
ary receiving  circuit.  Therefore 

xF_  _£' 

XI"  ~~  W 

and 

rF        E^ 

RF'  ~  W 

But 

E'      N' 

E"  =  W 

and 

/'  _  N" 

T»~  N' 

so  that 


(62) 

'  •»  r  */    I    ^fi. 


That  is,  a  transformer  supplying  current  from  its  secondary  to 
a  circuit  of  resistance  R  and  reactance  X,  takes  from  the  mains 
the  same  current  at  the  same  phase  lag  as  would  be  taken  by  a 

(  N1  V  I  N1  V 

circuit  of  resistance  (  -™  )  R  and  of  reactance  (  -^/>  )  ^  con- 
nected directly  to  the  mains. 

99.  Maximum  core  flux. — When  the  electromotive  force  acting 
on  the  primary  of  a  transformer  is  harmonic  there  is  a  simple 
and  important  relation  between  Ef ,  «,  A77,  and  the  maximum 
value  of  the  core  flux  $.  Let  ef  be  the  instantaneous  value  of 


THE   TRANSFORMER.  H9 

the  primary  electromotive  force.     Since  e'  is   assumed  to  be 

harmonic  we  have 

e'  =  &  sin  at  (i) 

Let  <f>  be  the  instantaneous  flux  through  the  core.  Then 
d$ldt  is  the  instantaneous  value  of  the  electromotive  force  in- 
duced in  each  turn  of  wire,  so  that 


or 

w  =f>sina*  (ji> 

Therefore 

j?r 
<f  =  —rj  cos  at  -f  a  constant  (iii) 


The  constant  of  integration  is  known  to  be  zero  inasmuch  as 
during  the  reversals  of  magnetization  of  the  core  the  flux  passes 
through  positive  and  negative  values  alike.  Therefore  equation 
(iii)  becomes 


The  coefficient  J^'/aN'  is  the  maximum  value  reached  by  <f 
since  the  maximum  value  of  cos  o*t  is  unity.  Therefore,  repre 
senting  the  maximum  value  of  the  core  flux  by  $,  we  have 

E' 


or,  since  the  maximum  value  J^'  of  the  primary  electromotive 
force  is  equal  to  \/2  times  its  effective  value  E'  ,  we  have 


100.  Transformer  losses.  —  The  power  output  of  a  transformer 
is  less  than  its  power  intake  because  of  the  losses  in  the  trans- 
former. These  losses  are  :  (a)  The  iron  or  core  losses  due  to 


150  ELEMENTS   OF   ALTERNATING   CURRENTS. 

eddy  currents  and  hysteresis  ;  and  (b)  the  copper  losses  due  to 
the  resistances  of  the  primary  and  secondary  coils. 

The  iron  losses  are  practically  the  same  in  amount  at  all  loads, 
and  they  depend  upon  the  frequency  and  range  of  the  flux  den- 
sity B,  upon  the  quality  and  volume  of  the  iron,  and  upon  the 
thickness  of  the  laminations. 

The  hysteresis  loss  in  watts  is 

WK*=aVf&*  (64) 

where/  is  the  frequency  in  cycles  per  second,  B  is  the  maxi- 
mum flux  density  in  lines  per  square  centimeter,  Fis  the  volume 
of  the  iron  in  cubic  centimeters,  and  a  is  a  constant  depending 
upon  the  magnetic  quality  of  the  iron.  For  annealed  refined 
wrought  iron  the  value  of  a  is  about  3  x  io~10. 
The  eddy  current  loss  in  watts  is  : 


2  (65) 

where  /  is  the  thickness  of  the  laminations  in  centimeters,  and 
b  is  a  constant  depending  upon  the  specific  electrical  resistance 
of  the  iron.  For  ordinary  iron  the  value  of  b  is  about  1  .6  x  io~  n. 
Insufficient  insulation  of  laminations  causes  excessive  eddy  cur- 
rent loss. 

Remark.  —  Equations  (64)  ind  (65)  maybe  used  for  calculating 
the  hysteresis  and  eddy  current  losses  in  any  mass  of  laminated 
iron  subjected  to  periodic  magnetization,  such  as  alternator  arma- 
tures and  the  rotor  and  stator  iron  in  an  induction  motor. 

The  copper  loss  is  : 

Wc  =  RT*  +  R"Im  (66) 

This  loss  is  nearly  zero  when  the  transformer  is  not  loaded  ;  it 
increases  with  the  square  of  the  current,  and  becomes  excessive 
when  the  transformer  is  greatly  overloaded. 

101.  Efficiency  of  transformers.  —  The  ratio  power  outputs- 
power  intake  is  called  the  efficiency  of  a  transformer.  The  ac- 


THE   TRANSFORMER.  I51 

companying    table    shows    the    full-load    efficiencies   of  various 
sized  transformers  of  a  recent  type. 

TABLE  OF  TRANSFORMER  EFFICIENCIES. 

OUTPUT  PER  CENT  EFFICIENCY 

KILOWATTS.  FULL  LOAD. 


Q"v7 

s 

•     • 

q6.2 

q6.4 

s 

p6.e 

s 

6                          

06.- 

s 

Q6.J 

& 

8                                     .   .   .   . 

q6.J 

;s 

q6.c 

» 

p6.( 

?5 

97- 

E 

The  efficiency  of  a  given       F" 

transformer  is  very  low  when       k 

the    output  is   small  ;    it  m-    '•  p 

|y 

s 

«x,^ 

creases    as    the    output    in-    -?^- 

f 

creases,  reaches  a  maximum,     $     / 

—  —J    fa11c   r»ff"   acrPITl    wTlPTI   the      .7      / 

nnfrmt  is  verv  pTeat      This    .<J  t- 

falling  off  of  efficiency  when    -s  Li- 

the output  is  preal-  is  due  to     J  J 

the  great  increase  of  copper    Jl 

losses.     Fie.  IIQ  shows  the    J| 

efficiency  of  a  transformer  at      V 

various  loads. 

fra, 

i/on 

*of 

A' 

'/« 

£ 

Calculation  of  efficiency.  — 
The  transformer  output  (non- 

\ 

\ 

7ig. 

i 

19. 

I 

/$ 

inductive  receiving  circuit)  is  E" I" .     The  internal  loss  is  Wh  -f 
W  4-  W   so  that  the  intake  is  E"  I"  -f  W,  +  Wf+  W,  and  the 

e     '  c' 

efficiency  is : 

T"  -L.  W  JL.  W  _L  ur  VU7) 


77  = 


All-day  efficiency. — Usually  a  transformer  is  connected  to  the 


152  ELEMENTS   OF   ALTERNATING   CURRENTS. 

mains  continuously,  and  current  is  taken  from  the  secondary  for 
a  few  hours  only,  each  day.  In  this  case  the  iron  loss  is  inces- 
sant and  the  copper  loss  is  intermittent.  The  total  work  given 
to  the  transformer  during  the  day  may  greatly  exceed  the  total 
work  given  out  by  it,  especially  if  the  incessant  iron  losses  are 
not  reduced  to  as  low  a  value  as  possible.  The  ratio  total  work 
given  out  by  the  transformer  -7-  total  work  received  by  the  trans- 
former during  the  day  is  called  the  all-day  efficiency  of  the  trans- 
former. 

102.  Practical  and  ultimate  limits  of  output  of  a  transformer. — 

When  the  secondary  current  of  a  transformer  is  increased  the 
secondary  electromotive  force  generally  drops  off,  and  the  power 
output  increases  with  the  current  and  reaches  a  maximum  as  in 
the  case  of  the  alternator.  This  maximum  power  output  is  the 
ultimate  limit  of  output  of  the  transformer.  Practically  the  out- 
put of  a  transformer  is  limited  to  a  much  smaller  value  than  this 
maximum  output  because  of  the  necessity  of  cool  running,  be- 
cause in  most  cases  it  is  necessary  that  the  secondary  electro- 
motive force  be  nearly  constant,  and  because  the  efficiency  of  a 
transformer  is  low  at  excessive  outputs. 

Small  transformers  have  relatively  large  radiating  surfaces  and 
in  such  transformers  the  requirements  of  close  regulation,  as  a 
rule,  determine  the  allowable  output. 

Large  transformers  have  relatively  small  radiating  surfaces  and 
their  allowable  output  is  limited  by  the  permissible  rise  in  tem- 
perature. Very  large  transformers  are  usually  provided  with  air 
passages  through  which  air  is  made  to  circulate  by  a  fan.  Some- 
times transformers  are  submerged  in  oil,  which,  by  convection, 
carries  heat  from  the  transformer  to  the  containing  case,  where 
it  is  radiated. 

Large  transformers  are  much  more  efficient,  under  full  load, 
than  small  ones,  and  give  closer  regulation. 

103.  Rating  of  transformers. — A  transformer  is  rated  accord- 
ing to  the  power  it  can  deliver  steadily  to  a  non-inductive  receiv- 


THE   TRANSFORMER.  153 

ing  circuit  without  undue  heating  ;  and  the  ratio  of  transforma- 
tion, together  with  a  specification  of  the  frequency  and  effective 
value  of  the  primary  electromotive  force  to  which  the  transformer 
is  adapted,  are  given. 

The  rating  of  a  transformer  is  by  no  means  rigid.  Thus,  if  a 
transformer  is  used  to  give  more  than  its  rated  output  it  will 
become  somewhat  more  heated  by  the  internal  losses  and  its 
regulation  will  not  be  so  close.  If  a  transformer  is  used  for  a 
primary  electromotive  force  greater  than  its  rated  electromotive 
force  or  for  a  frequency  lower  than  its  rated  frequency,  the  range 
of  flux  density  B  in  the  core  will  be  increased,  which  will  increase 
the  core  losses.  Some  manufacturers  rate  their  transformers 
generously,  so  that  they  may  be  greatly  overloaded  or  used  with 
greatly  increased  primary  electromotive  force  or  decreased  fre- 
quency without  difficulty. 

104.  Outline  of  transformer  design. — A  transformer  is  usually 
designed  to  take  current  from  mains  at  a  prescribed  electromotive 
force  and  frequency,  and  to  deliver  current  at  a  prescribed  electro- 
motive force  to  a  receiving  circuit.  The  transformer  must  be  so 
proportioned  and  of  such  size  as  to  deliver  the  prescribed  amount 
of  current  steadily  without  undue  heating  and  without  any  great 
variation  of  its  secondary  electromotive  force  from  zero  to  full  load. 

In  the  designing  of  a  transformer  there  is  but  one  condition 
which  must  be  precisely  met,  namely,  the  ratio  of  primary  to  sec- 
ondary turns  must  be  equal  to  the  ratio  of  the  prescribed  primary 
and  secondary  electromotive  forces.  All  other  points  in  design 
are  to  a  great  extent  matters  of  choice  guided  in  a  general  way 
by  experience. 

The  accompanying  table  gives  magnetic  flux  densities  which 
are  usually  employed  in  transformer  cores. 

The  allowable  temperature  rise  varies  greatly  with  different 
makers,  the  extent  of  radiating  surface  required  per  watt  of  loss 
per  degree  rise  of  temperature  varies  between  extremely  wide 
limits,  and  no  simple  rule  can  be  given  covering  this  matter. 


154 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


MAGNETIC  DENSITIES  B  FOR  TRANSFORMER  CORES. 


FREQUENCY. 

SMALL 
TRANSFORMERS. 

MEDIUM  SIZE 
TRANSFORMERS. 

LARGE 
TRANSFORMERS. 

25 

7500 

6750 

6000 

40 

6500 

5750 

5000 

60 

5OOO 

4750 

4500 

80 

4500 

4250 

4000 

100 

4000 

3750 

35°° 

120 

3500 

3250 

3000 

Given  the  required  power  output  *  of  a  transformer,  the  value 
and  frequency  of  the  primary  electromotive  force  and  the  value 
of  the  secondary  electromotive  force,  the  design  of  the  trans- 
former is  conveniently  determined  as  follows  : 

Find  from  the  table  the  efficiency  which  can  probably  be  at- 
tained, and  calculate  the  total  transformer  loss  at  full  load.  Of 
this  total  loss  about  half  should  be  iron  loss  and  half  copper 
loss.f  The  total  iron  loss  is  : 

W.  =  a  VfB1-6  +  b  Vf'WB2  (68) 

according  to  equations  (64)  and  (65). 

Having  decided  upon  maximum  flux  density  B  (see  accom- 
panying table)  and  upon  thickness  J  of  laminations  /,  equation 
(68)  gives  the  volume  Fof  iron  to  be  used  in  the  transformer 
core.  The  core  may  be  made  of  the  type  shown  in  Fig.  1 20  or 
of  the  type  shown  in  Fig.  121.  The  proportions  (relative  dimen- 
sions) indicated  in  Figs.  1 20  and  1 2 1  will  be  found  to  give  satis- 
factory results  although  the  form  of  the  core  may  be  considerably 
modified  without  greatly  affecting  the  action  of  the  transformer. 
In  fact,  it  is  usually  necessary  to  modify  the  core  slightly  after 
the  coils  have  been  designed. 

*  Rated  output  is  the  output  which  the  transformer  can  deliver  satisfactorily  to  a 
non-inductive  circuit. 

|  If  the  transformer  is  to  be  connected  to  the  mains  all  day,  but  is  to  deliver  cur- 
rent only  four  hours  per  day,  for  example,  then  the  iron  loss  during  24  hours  should 
be  about  equal  to  the  copper  loss  during  four  hours,  or  under  the  full  load  the  copper 
loss  should  be  several  times  as  great  as  the  iron  loss. 

|  12  to  1 6  thousandths  of  an  inch  is  the  thickness  usually  employed. 


THE   TRANSFORMER. 


155 


! 

B 

*o 

fjfll 

*lia 

«-  a  -*• 

8 
c* 

1 

«*« 

> 

Fig.  120. 


The  maximum  core  flux  $  is  equal  to  the  product  of  the  sec- 
tional area  of  the  magnetic  circuit  (where  it  passes  through  the 


^rh« 


5 


Fig.  121. 


coils)  into  the  maximum  flux  density  B.     Then  equation  (63) 
determines  the  primary  turns,  namely, 


I0 


or 


E'  io8 


"  -4.44*/  (69) 

The  number  of  secondary  turns  is  then  determined  by  equation 
(61)  namely, 

N"  =^rNf  (70) 


56 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


From  the  provisionally  designed  core  the  mean  length  of  a 
turn  of  primary  and  of  secondary  coils  may  be  determined  which 
together  with  Nf  and  N"  gives  the  total  lengths  of  wire  in 
primary  and  secondary.  The  size  of  this  wire  is  then  easily 
chosen  so  that  R '/'2  and  R"I"2  may  be  each  equal  to  half  the 
full-load  copper  loss. 

The  size  of  wires  being  thus  determined  the  space  necessary 
for  the  coils  and  insulation  can  be  estimated.  If  the  provisionally 
designed  core  gives  more  or  less  space  than  is  required  for  the 
coils  its  dimensions  may  be  altered  to  suit. 

TRANSFORMER  CONNECTIONS. 

105.  Simple  connection.  In  parallel.  In  series. — When  used 
to  supply  current  to  lamps  or  motors  from  constant  potential 


recieving\  \  recieviruj\ 

circuit  circuit 

Fig.  122. 

mains  the  primary  of  the  transformer  is  connected  to  the  mains 
and  the  secondary  of  the  transformer  is  connected  to  the  terminals 
of  the  receiving  circuit.  When  a  number  of  receiving  circuits 


L-* — I 


I        w          I 


Fig.  123. 


are  supplied  through  separate  transformers  the  primaries  of  the 
transformers  are  connected  in  parallel,  as  shown  in  Fig.  122. 


THE   TRANSFORMER. 


157 


When  current  is  supplied  through  transformers  to  a  number  of 
arc  lamps  from  a  constant  current  alternator  the  transformer  pri- 
maries are  connected  in  series  and  the  lamps  are  connected  to  the 
respective  secondaries,  as  shown  in  Fig.  123.  This  arrangement 
is  seldom  employed. 

106.  Transformers  with  divided  coils. — Alternators  for  isolated 
lighting  plants  give  usually  I,OOO  or  2,000  volts  electromotive 

main 


main 


IOOV- 


=3 


-IOOV- 


-200V- 


main 


main 


lamps  la  mi 

Fig.  124. 

force  and  the  standard  electromotive  forces  for  incandescent  lamps 

are  5  5  and  1 1  o  volts.     Transformers  are  frequently  made  with 

two    primary   coils,   which 

may  be  connected  in  series    •— 

for  2,000  volts  or  in  parallel 

for    1,000  volts,  and  with   — 

two  secondary  coils,  which 

may  be  connected  in  series 

to    give    1 1  o   volts    or   in  ' 

parallel  to  give  55  volts. 

Transformers  for  supply- 
ing current  for  testing  pur- 
poses are  frequently  made 

with  a  number  of  secondary  coils,  which  may  be  connected  to  give 
high  or  low  electromotive  forces  as  desired. 


158 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


Transformers  for  supplying  current  to  the  Edison  three-wire 
system. — For  this  purpose  two  similar  transformers  may  be  used 
as  shown  in  Fig.  1 24,  or  a  single  transformer  with  two  secondary 
coils  may  be  used  as  shown  in  Fig.  125. 

107.  The  autotransformer. — The  two  coils  of  a  transformer 
may  be  connected  in  series  between  supply  mains  and  the  receiv- 
ing circuit  connected  to  the  terminals  of  either  coil  as  shown  in 
Fig.  1 26 ;  or  either  coil  of  the  transformer  may  be  connected  to 
the  supply  mains  and  current  delivered  to  the  receiving  circuit 
through  the  two  coils  in  series  as  shown  in  Fig.  127.  A  trans- 


fiupply  mains 


rtfTftnr 


/Service'- 
wains 


$  apply 
mains 


^jQ 


Service  mains 


Fig.  126. 


Fig.  127. 


former  arranged  in  this  way  is  called  an  autotransformer.  This 
arrangement  of  a  transformer  is  especially  advantageous  when 
the  ratio  of  supply  electromotive  force  to  receiver  electromotive 
force  is  nearly  unity. 

Electromotive  force  relations. — Let  P  and  vS  be  the  electro- 
motive forces  induced  in  the  respective  coils  by  the  alternations 
of  the  core  flux.  These  electromotive  forces  are  proportional  to 
the  number  of  turns  of  wire  in  the  respective  coils. 

From  Fig.  126  we  have 


THE   TRANSFORMER.  159 

so  that 

E,      P±S  . 

4"^~  W 

The  sign  is  -f  or  —  according  as  the  coils  are  so  connected  that 
the  induced  electromotive  forces  are  in  the  same  direction  or  in 
opposite  directions  respectively  between  a  and  b.  If  the  receiv- 
ing circuit  is  connected  to  P  in  Fig.  1  26  we  have 


,  •         CO 

From  Fig.  127  we  have 


so  that 


^2 

If  the  supply  mains  are  connected  to  5  in  Fig.  127  we  have 


(V) 

In  equations  (a)  and  (fr)  P  and  5  may  be  simply  the  numbers 
of  turns  of  wire  in  the  respective  coils. 

Current  relations. — The  magnetizing  actions  of  the  currents  in 
the  coils  P  and  5  are  at  each  instant  equal  and  opposite  (core  re- 
luctance zero).  Therefore,  the  currents  in  coils  P  and  .S  are  in- 
versely proportional  to  the  number  of  turns,  so  that  we  may  rep- 
resent by  P  the  current  in  the  coil  S,  and  by  5  the  current  in 
the  coil  P.  These  currents  may  be  made  to  flow  in  the  same  or 
in  opposite  directions  through  the  circuit  PS  by  reversing  the 
connections  of  one  coil,  and  the  current  in  the  main  which  con- 
nects to  the  middle  terminal  of  the  two  coils  is  P±  S,  according 
as  the  currents  flow  at  a  given  instant  in  opposite  directions,  or  in 
the  same  direction  with  reference  to  the  circuit  PS. 

Let  7X  be  the  current  supplied  to  the  autotransformer  and  /2 


160  ELEMENTS   OF   ALTERNATING   CURRENTS. 

the  current  delivered  to   the   receiving  circuit.     Then,   on   the 
basis  of  the  above  considerations,  we  have  from  Fig.  126 


and 
so  that 


// 


If  the  receiving  circuit  is  connected  to  P  in  Fig.  1  26,  we  have 

* 


- 

, 


From  Fig  127  we  have 


so  that 


If  the  supply  mains  are  connected  to  5  in  Fig.  127,  we  have 

K) 

2  • 

High  duty  of  autotransformer.  —  Only  a  portion  of  the  power 
delivered  by  an  autotransformer  is  transformed  through  the  med- 
ium of  the  iron  core  from  one  coil  to  the  other.  The  remainder 
is  supplied  to  the  receiving  circuit  direct  by  virtue  of  the  series 
connection.  Thus,  to  take  an  extreme  case,  if  an  autotransformer 
takes  1  1  amperes  from  I  oo-volt  mains  and  delivers  I  o  amperes 
at  1  10  volts,  of  the  total  1,100  watts  only  100  watts  are  actually 
transformed,  as  may  readily  be  shown  by  a  careful  scrutiny  of 
the  above  discussion.  That  is,  the  transformer,  so  far  as  the  size 
of  its  wires  and  its  induced  electromotive  forces  are  concerned,  is 
a  transformer  which  would  be  rated  as  a  loo-watt  transformer. 

Remark.  —  It  is  not  allowable  in  practice  to  connect  transform- 
ers as  autotransformers  since  this  manner  of  connection  involves 


THE   TRANSFORMER. 


161 


the  connection  of  low  electromotive  force  service  mains  to  the 
high  electromotive  force  transmission  mains,  and  a  ground  con- 
nection on  the  transmission  mains  becomes  dangerous. 

108.  Arrangements  of  transformers  in  polyphase  systems  (with- 
out changing  the  number  of  phases}. — In  general,  step-up  and 
step-down  transformation  of  polyphase  currents  is  accomplished 
by  using  an  independent  transformer  for  each  phase.  In  case  of 
a  three-wire  three-phase  system,  the  primaries  of  the  three  sepa- 
rate transformers  may  be  A-connected  or  Y-connected  to  the 
supply  mains,  and  the  three  secondaries  may  be  A-connected  or 
Y-connected  to  the  service  mains. 

In  a  three-wire  three-phase  system,  transformation  may  be  ac- 
complished by  connecting  two  transformers  exactly  as  they  would 
be  connected  for  a  three-wire  two-pltase  system,  as  shown  in  Fig. 


apply  main  1 


jServiee    main  1 


main  2 


1  ervice  in  am 


main  3 


Jtfervice  main  3 


Fig.  128. 


128.  This  arrangement  is  frequently  used  in  practice.  When 
the  secondaries  are  properly  connected  to  the  service  mains  the 
electromotive  forces  ab,  be,  and  ca  are  equal  in  value  and  120° 
apart  in  phase.  The  reversing  of  the  connections  of  the  secon- 
dary of  the  one  or  the  other  transformer  gives  the  following  elec- 
tromotive forces  between  the  service  mains,  namely,  ab  and  be  are 
equal  in  value  and  60°  apart  in  phase,  and  ca  is  \/3  times  as 
large  as  ab  and  be  and  midway  between  them  in  phase. 


162 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


109.  Two-phase    three-phase   transformers. — Transformers  are 
frequently  used   to  transform  from  two-phase  supply  mains  to 
three-phase  service  mains  or  vice  versa.     The  principle  involved 
in  this  kind  of  transformation   is  best  brought  out  by  consider- 
ing the  following  prelimi- 
nary problem,  namely  : 

To  produce  an  electromo- 
tive force  of  any  specified 
value  and  phase. — Let  A 
1  and  B,  Fig.  1 29,  be  the  two 

electromotive  forces  of  a 
—4**2$  two-phase  dynamo  and  let 
it  be  required  to  produce 
an  electromotive  force  El 
of  any  given  value  and 
phase.  The  component  of 
El  parallel  to  A  is  E^  sin  ft 


I    J^  JL 

Fig.  129. 

parallel  to  B  is  Ev  cos  /3. 


and  the  component  of  E^ 
Fig.  130  shows  two  distinct  trans- 
formers with  similar  primary  coils,  one  connected  to  phase  At  the 
other  to  phase  B.  A  secondary  a  may  be  wound  upon  the  one 
transformer  to  give  the  component  E^  sin  /3  ;  and  a  secondary  b 
may  be  wound  upon  the 
other  transformer  to  give  the 
other  component  El  cos  /3. 
These  two  secondaries  when 
connected  in  series  give  the 
desired  electromotive  force 
Ev  Similarly  any  other  elec- 
tromotive force,  such  as 

^  »."**•  FiS-    130' 

E2   or  E#    Fig.    129,    may 

be  produced  by  a  pair  of  properly  proportioned  secondary  coils. 

The  two-phase  three-phase  transformer  consists  of  two  distinct 

transformers  A  and  B,  Fig.    130,  wound  with   similar   primary 

coils  to  which  the  two-phase  electromotive  forces  are  connected 


THE   TRANSFORMER. 


Each  of  the  three-phase  electromotive  forces  is  (in  general)  gen- 
erated in  a  pair  of  secondary  coils,  one  on  each  transformer. 
Such  a  pair  of  coils  constitutes  a  three-phase  unit.  The  three 
units  may  be  connected  according  to  the  A  scheme  or  Y  scheme 
to  the  service  mains.  In  the  first  case  the  electromotive  forces 
between  the  three-phase  mains  are  the  electromotive  forces  pro- 
duced in  the  respective  pairs  of  coils.  In  the  second  case  the 
electromotive  forces  between  the 
mains  are  related  to  the  electromo- 
tive forces  generated  by  the  respec- 
tive pairs  of  coils  as  explained  in 
Article  73.  Such  a  transformer 
transforms  equally  well  from  three- 
phase  to  two-phase  or  from  two- 
phase  to  three-phase. 

The  Scott  transformer. — To  un- 
derstand the  Scott  transformer, 
which  is  the  simplest  kind  of  two- 
phase  three-phase  transformer,  it  is  helpful  to  consider  the  most 
general  possible  type  of  two-phase  three-phase  transformer  as 
follows :  Consider  any  point  O,  Fig.  131,  from  which  are  drawn 
three  lines,  ay  b  and  r,  terminating  at  the  corners  of  an  equilateral 

triangle  pqr.  If  three  pairs 
of  coils  are  arranged  on  the 
cores  A  and  B,  Fig.  130, 
one  pair  to  give  the  electro- 
motive force  a,  another  pair 
to  give  the  electromotive 
force  b,  and  the  third  pair  to 
give  the  electromotive  force 

1  2  3  cy  Fig.  131,  then  these  three 

pairs  of  coils  Y-connected 

to  three  mains  would  give  the  symmetrical  three-phase  electromo- 
tive forces  mmm,  Fig.  131.  The  three-phase  units  of  this  general 
type  of  two-phase  three-phase  transformer  cannot  be  A-connected. 


Fig.  131. 


a        b 


c 


164 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


Scott's  transformer  consists  of  two  cores  with  similar  primaries 

A  and  B,  Fig.  132.     These  two  primaries  are  connected  to  the 

two-phase  mains.      One  core  has  two  similar  secondaries  a  and  b, 

and  the  other  core  has  a  single  secondary  c  having  \/3  times  as 

q  many  turns'  as  a  or  b. 

The  coils  a,  b  and  c  are 
Y  -  connected  to  the 
three-phase  mains  I,  2 
and  3,  as  shown.  The 
point  0,  Fig.  131,  lies, 

Hr     *  "~»"   for  Scott's  transformer, 

midway  between  the 
In  this  figure  a,  b  and  c 


P 


A 


m 


Fig.  133. 

points  p  and  r  as  shown  in  Fig.  133. 
represent  the  respective  electromotive  forces  induced  in  the  sec- 
ondary coils  a,  b  and  c, 
Fig.  132.  The  two-phase 
electromotive  forces    A 

and  B  are  parallel  to  a      /  winding  A 

and  to  c  respectively,  as   2 
shown.      Scott's    trans- 
former cannot  be  A-con- 
nected. 

main 


Tnain          [ 


Fig.  134. 

110.  The  monocyclic  sys- 
tem.— The  monocyclic  gen- 
erator of  the  General  Elec- 
tric Company  is  a  polyphase 
dynamo  not  strictly  to  be 
called  two-phase  or  three- 
phase.  It  is  employed  in  sta- 
tions where  a  small  portion 
of  the  output  is  used  for 
motors  and  a  large  portion  for  lighting.  The  armature  winding 
of  the  monocyclic  generator  is  essentially  a  two-phase  winding. 
The  A  winding  has  four  times  as  many  conductors  as  the  ^winding, 


A'      [I—— .  i 


Fig.  135. 


THE  TRANSFORMER.  165 

and  one  end  of  the  B  winding  is  connected  to  the  middle  point 
of  the  A  winding,  as  shown  in  Fig.  1 34.  The  three  collecting 
rings  are  indicated  by  I,  2  and  3.  Main  3  is  called  the  teaser. 

Lamps,  or  transformers  feeding  lamps,  are  connected  to  mains 
I  and  2,  and  two  similar  transformers  connected,  as  shown  in  Fig. 
135,  are  used  to  supply  three-phase  currents  to  induction  motors. 


PROBLEMS. 

8 1.  The  primary  coil  of  a  transformer  has  60  turns  of  wire,  it 
takes  750  amperes  from  an  alternator  at  1 10  volts  and  steps  up 
to   20,000   volts.      How   many  turns  of  wire  are  there  in  the 
secondary  coil  ?     A  usual  allowance  in  transformers  is   500  cir- 
cular mils  of  sectional  area  of  wire  for  each  ampere  of  current. 
Find  size  of  primary  wire  and  of  secondary  wire  of  the  above 
transformer.     Ans.  10,909  turns,  375,000  circular  mils,  2,062.5 
circular  mils. 

82.  A  transformer  has  800  turns  of  wire  on  the  primary  coil, 
40  mils  in  diameter,  and  80  turns  on  the  secondary,  secondary 
wire  being  ten  times  as  large  in  sectional  area.     The  sectional 
area  of  the  magnetic  circuit  of  the  iron  core  is  1 2  square  inches. 
Allowing  500  circular  mils  of  conductor  per  ampere,  and  allow- 
ing a  maximum  flux  density  of  4,000  lines  per  square  centimeter, 
calculate  the  electromotive  force,  current  and  power  ratings  of 
the  transformer  at  60  cycles  per  second.     Ans.  656  volts,  32 
amperes,  2.099  kilowatts. 

83.  A  given  transformer  is  rated  at  5  kilowatts  and  is  designed 
to  take  current  from  i,ioo-volt  mains  at  a  frequency  of  60  cycles 
per  second.     Under  these  conditions   Wh,    We  and   Wc  will    be 
called  normal. 

(a)  The  transformer  is  used  at  6  kilowatts  output  at  the  rated 
electromotive  force  and  frequency.  Find  IVC  in  terms  of 
normal. 


1 66  ELEMENTS   OF   ALTERNATING   CURRENTS. 

(ft)  The  transformer  is  used  at  rated  electromotive  force  but  at 
a  frequency  of  75  cycles  per  second.  Find  Wh  and  We  each  in 
terms  of  normal. 

(c)  The  transformer  is  used  at  rated  frequency  but  with  primary 
electromotive  force  of  1,500  volts.     Find    Wh  and   We  each  in 
terms  of  normal. 

(d)  The  transformer  is  used  on  primary  electromotive  force  of 
1,500  volts.      Find  f  for  which  Wh  is  normal.     Show  that  with 
given  E,  We  is  independent  of  frequency. 

(e)  With  primary  electromotive  force  of  1,500  volts  what  load 
would  give  normal  Wc  ? 

Ans.  (a)  1.44  x  normal  Wc ;  (ft)  0.87  x  normal  Wh,  no  change 
in  We\  (c)  164  x  normal  Wh,  186  x  normal  W9\  (d]  36.5  cycles  per 
second. 

84.  A  transformer  has  1,300  turns  of  wire  in  its  primary  coil 
and   130  turns  in  its  secondary  coil.     The  primary  coil  takes 
current  from  i,ioo-volt  mains  and  the  secondary  coil  delivers 
200  amperes  to  a  receiving  circuit  of  which  the  power  factor 
is  0.85.     What  is  the  equivalent  primary  resistance  and  reactance 
of  the  secondary  receiving  circuit?     Ans.  r—  21.67  ohms,  x  = 
13.44  ohms. 

85.  The  sectional  area  of  the  core  of  the  transformer  in  prob- 
lem   8 1   is    161   square  centimeters.      Find    the  maximum  core 
flux  and  maximum  flux  density  at  a  frequency  of  60  cycles  per 
second.     Ans.  $  =  691,000,  B  =  4,250. 

86.  A  bundle  of  iron  wires  of  which  the  total  sectional  area  is 
1 2. 6  square  centimeters  is  to  be  magnetized  by  alternating  current 
taken  from  I33~cycle  no-volt  mains  so  that  the  maximum  flux 
density  in  the  core  may  be  4,500  lines  per  square  centimeter. 
How  many  turns  of  wire  are  required  ?     How  many  turns  of  wire 
would  be  required  at  60  cycles  per  second,  other  things  remain- 
ing the  same?     Ans.  330  turns,  728  turns. 

87.  An  electromotive  force  of  no  volts,  from  a  battery  for 
example,  is  applied  during  ^^  second  intervals  in  reversed  direc- 


THE   TRANSFORMER.  167 

tions  to  a  coil  of  I  oo  turns  of  wire  wound  on  an  iron  core.  Plot  the 
curve  representing  the  above  electromotive  force  and,  neglecting 
the  resistance  of  the  coil,  plot  the  curve  representing  the  core 
flux  produced.  The  iron  core  is  supposed  to  have  a  constant 
magnetic  reluctance  of  0.00014.  Plot  the  curve  representing  the 
magnetizing  current. 

A  secondary  coil  of  50  turns  is  wound  on  the  above  core. 
This  secondary  coil  supplies  current  to  a  non-inductive  receiving 
circuit  having  200  ohms  resistance.  Plot  curves  representing 
(a)  secondary  electromotive  force,  (b)  secondary  current,  and  (c) 
total  primary  current.  Resistance  of  secondary  coil  is  to  be 
neglected. 

88.  A  transformer  has  its  primary  in  two  sections  which  can 
be  thrown  in  series  or  in  parallel  at  will.     This  primary  takes 
current  through  a  rheostat  of  resistance  R' ',  and  the  secondary 
supplies  current  to  a  non-inductive  receiving  circuit  of  resistance 
R".     Find  the  relation  between  R' ,  R",  Nr  (total),  and  N"  for 
which  the  secondary  current  is  the  same  whether  the  primary 

(~  N*  Y 
AT"  )  ^ ' ' 

89.  An  ordinary  transformer,  rated  at  one  kilowatt,  100  volts 
primary,  and   10  volts  secondary,  is  connected  up  as  an  auto- 
transformer  to  transform  from  100  volts  to  no  volts.     What  is 
the  full  load  rating  of  the  transformer  so  used,  what  is  the  full 
load  output  of  current,  and  what  is  the  full  load  intake  of  cur- 
rent?    What  is  the  full  load  current  in  the  loo-volt  coil,  and 
what  is  the  full  load  current  in  the  secondary  coil  ?     Ans.   (a)  1 1 
kilowatts,  (b)  loo  amperes,  (c)  1 10  amperes,  (*/)  10  amperes,  (e) 
100  amperes. 

Assuming  the  coils  to  be  as  right-handed  helices  in  one  layer 
on  an  iron  core,  give  a  diagram  of  the  connections. 

90.  The  primary  coils  of  two  transformers    have  each   560 
turns  of  wire  and  they  are  connected  to  two-phase  mains,  the 
electromotive  force  of  each   phase  being  800  volts.      Calculate 


1 68  ELEMENTS   OF  ALTERNATING   CURRENTS. 

the  turns  of  wire  in  each  of  two  secondary  coils  (one  on  each 
transformer)  so  that  these  coils  when  connected  in  series  give  an 
electromotive  force  of  400  volts,  30°  ahead  of  one  of  the  two- 
phase  electromotive  forces.  Ans.  140  turns,  242.5  turns. 

91.  A  Scott  transformer  is  to  transform  from  1,000  volts,  60 
cycles,  two-phase,  to  100  volts  three-phase.     The  cross  section 
of  each  iron  core  is  75  square  centimeters.     Find  the  number  of 
turns  of  wire  in  each  primary,  and  in  each  secondary  coil,  allow- 
ing a  maximum  flux  density  of  3,500  lines  per  square  centimeter. 
Ans.  N'=  1,430,  0  =  71.5,  £=71.5,  c=  124. 

92.  Three  similar   1,000  to   loo-volt  transformers  have  their 
i,ooo-volt  coils  A-connected   to  three-phase   i,ooo-volt  mains. 
The  secondaries  are  Y-connected  to  service  mains.     Give  a  dia- 
gram of  the  connections  and  find  the  electromotive  force  between 
the  pairs  of  service  mains.     Ans.  173  volts. 


CHAPTER   XI. 
THE   TRANSFORMER. 

(  Continued. ) 

111,  The  actual  transformer  and  the  ideal  transformer. — The 

discussion  given  in  Articles  95  to  99  ignores  the  resistances  R' 
and  R"  of  the  transformer  coils,  it  takes  but  little  account  of  the 
magnetizing  current  m,  which  depends  upon  eddy  currents  and 
magnetic  hysteresis  in  the  iron  core,  and  it  assumes  that  all  the 
lines  of  magnetic  flux  which  pass  through  one  coil  pass  through 
the  other  also.  A  transformer  which  would  meet  these  condi- 
tions would  be  an  ideal  transformer.  A  well-designed  trans- 
former operating  on  moderate  load  does  approximate  quite 
closely  to  the  ideal  transformer  in  its  action,  and  equations  (60) 
and  (63)  are  much  used  in  practical  calculations.  For  some  pur- 
poses, however,  it  is  desirable  to  consider  the  action  of  the  trans- 
former, taking  account  of  coil  resistances,  of  eddy  currents  and 
hysteresis,  and  of  the  fact  that  some  lines  of  magnetic  flux  pass 
through  one  coil  without  passing  through  the  other  (magnetic 
leakage).  The  present  chapter  is  devoted  to  this  discussion. 

The  effects  of  coil  resistances,  of  eddy  currents  and  hysteresis, 
and  of  magnetic  leakage  are  small.  Their  influence  on  each 
other  is  very  much  smaller  and  is  ignored  in  the  following  discus- 
sion. That  is,  the  effects  of  coil  resistances,  the  effects  of  eddy 
currents  and  hysteresis,  and  the  effects  of  magnetic  leakage  are 
considered  separately. 

We  shall  first  discuss  these  various  effects  with  the  help  of 
vector  diagrams  for  the  sake  of  clearness,  giving  aftenvards  the 
formulation  of  the  symbolic  equations  and  Steinmetz's  solution. 

169 


1 70  ELEMENTS   OF   ALTERNATING   CURRENTS. 

112,  The  magnetizing  current  of  a  transformer  is  not  harmonic, 
as  is  shown  below,  but  since  it  is  usually  small,  it  may,  for  the 
purpose  of  calculation,  be  replaced  by  an  equivalent  harmonic 
current  M,  of  which  the  component  parallel  to  E1  (the  power 
component)  is  Mp  ;  and  the  component  at  right  angles  to  E'  (the 
wattless  component)  is  Mw.  The  power  taken  from  the  mains  by 
the  magnetizing  current  is  E1 'Mp  and  this,  ignoring  coil  resist- 
ances, is  equal  to  the  total  core  loss,  We  -f  Wh  [see  equations  (64) 
and  (6  5)].  Therefore 

^-^  w 

The  wattless  component  Mu  of  the  magnetizing  current  reaches 
its  maximum  value  \/2  Mw  when  the  core  flux  is  at  its  maximum 
value  $,  and  inasmuch  as  this  component  of  the  magnetizing 
current  overcomes  the  magnetic  reluctance  of  the  core  we  have 


or 


in  which  G  is  the  magnetic  reluctance  of  the  transformer  core  cor- 
responding to  the  maximum  flux  $. 

Admittance  of  the  transformer  at  zero  output.  —  The  magnetizing 
current  is  the  current  that  flows  through  the  primary  coil  when 
the  secondary  current  is  zero,  and  the  actual  primary  current  al- 

N" 
ways  exceeds  the  ideal  primary  current,  -^  •  /",  by  the  amount 

of  the  magnetizing  current.  Therefore  the  effect  of  the  magne- 
tizing current  may  be  represented  in  an  ideal  transformer  by  con- 
necting in  parallel  with  the  primary  coil  a  circuit  through  which 
flows  a  current  equal  in  value  and  phase  to  the  magnetizing  cur- 
rent of  the  actual  transformer.  The  admittance  to  this  shunt 
circuit,  g^  —jblt  is  used  in  the  symbolic  solution  of  the  transformer 
problem.  The  values  of  g^  and  bl  are  determined  from  the  mag- 


THE   TRANSFORMER. 


I/I 


netizing  current  as  follows  :  The  component  of  M  parallel  to  Er 
is  g^E*  (==  Mp),  and  the  component  of  M  perpendicular  to  E'  is 
b^E'  (=  Mw).  See  equations  (52)  and  (53).  Therefore,  using 
the  values  of  Mp  and  Mu  from  equations  (a)  and  (&),  we  have 


and 


or  using  the  value  of  §  from  equation  (63)  we  have 

10  G 

0*  = 


(72) 


113.  Actual  value  of  the  part  m  of  the  primary  current. 

(a)  When  the  core  is  assumed  to  be  without  hysteresis. —  Let  the  ordinates  of  the 
curve  m<b,  Fig.  136,  represent  values  of  the  core  flux  <l>  produced  by  various  given  cur- 
rent strengths  m  in  the  primary  coil,  these  current  strengths  being  represented  by  the 
abscissas  of  the  curve  m$. 

When  the  primary  electromotive  force  is  harmonic  then  the  core  flux  4>  is  harmonic 
also,  and  90°  behind  Ef  in  phase,  according  to  equations  (ii)  and  (iii),  Article  99. 
Let  the  sine  curve  $, 


axis  of  lime,  andin 


Fig.  136. 

Fig.   1 36,  represent  the  value  of  4»  as  time  passes ;  time  as  abscissas  and  4>  as  ordi- 
nates. 

Then  the  curve  m  of  which  the  ordinates  represent  successive  instantaneous  values 
of  the  current  m  is  constructed  as  follows  :  Draw  the  ordinate  dp  and  the  abscissa 
ap.  Lay  off  dc  equal  to  ab  which  is  the  magnetizing  current  required  to  force  through 


1/2 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


the  core  the  flux  dp.     The  locus  of  the  point  c  is  the  required   curve.     The  figure 
shows  that  the  magnetizing  current  is  not  harmonic  although  it  is  wattless. 

(b)  When  the  hysteresis  is  taken  into  account. — Let  the  ordinates  of  the  curve  m$, 
Fig.  137,  represent  values  of  the  core  flux  produced  by  various  given  current  strengths 
in  the  primary  coil,  these  current  strengths  being  represented  by  the  abscissas  of  the 
curve  m$.  The  curve  m  of  which  the  ordinates  represent  the  successive  instantaneous 


Fig.  137. 

values  of  the  current  m  is  constructed  as  before  ;  the  ascending  branch  of  the  hystere- 
sis loop  m$  being  used  for  increasing  values  of  4>  and  the  descending  branch  for  de- 
creasing values  of  $. 

114.  Transformer  regulation.  Preliminary  statement  concerning 
the  effects  of  magnetic  leakage  and  of  resistances  of  primary  and 
secondary  coils  on  the  action  of  a  transformer. — In  the  ideal  trans- 
former the  whole  of  the  primary  electromotive  force  is  balanced 
by  the  opposite  electromotive  force  induced  in  the  primary  coil 
by  the  varying  magnetic  flux  which  passes  through  both  coils, 
and  the  whole  of  the  electromotive  force  induced  in  the  secon- 
dary coil  by  this  varying  flux  is  available  at  the  terminals  of  the 
secondary  coil. 

In  the  actual  transformer  a  portion  of  the  primary  electro- 
motive force  is  lost  in  overcoming  the  resistance  of  the  primary 
coil  and  a  portion  is  lost  in  balancing  the  electromotive  force 
which  is  induced  in  the  primary  coil  by  the  flux  which  passes 
through  the  primary  coil,  but  does  not  pass  through  the  secon- 
dary coil  (leakage  flux).  These  lost  portions  of  the  primary 
electromotive  force  are  proportional  to  the  primary  current,  so 


THE   TRANSFORMER.  173 

that  the  useful  part*  of  the  primary  electromotive  force  falls  short  f 
of  the  total  primary  electromotive  force  by  an  amount  which  is 
proportional  to  the  current. 

The  total  electromotive  force  induced  in  the  secondary  coil  is 
proportional  to  the  useful  part  o>i  the  primary  electromotive  force 
and  a  portion  of  the  total  secondary  electromotive  force  is  lost  in 
overcoming  the  resistance  of  the  secondary  coil.  This  lost  por- 
tion of  the  secondary  electromotive  force  is  proportional  to  the 
secondary  current  (or  to  primary  current,  since  the  ratio  of  the 
currents  is  constant). 

Therefore  the  effect  of  magnetic  leakage  and  of  coil  resistances 
is  to  make  the  electromotive  force  between  the  terminals  of  the 

N" 
secondary  coil  fall  short  f  of  its  ideal  value -v^-  •  E'  by  an  amount 

which  is  proportional  to  the  current. 

This  falling  off  of  secondary  electromotive  force  with  increasing 
current  is  of  practical  importance,  inasmuch  as  most  receiving 
apparatus  must  be  supplied  with  current  at  approximately  con- 
stant electromotive  force.  A  transformer  of  which  the  secondary 
electromotive  force  falls  off  but  little  with  increase  of  current  is 
said  to  have  good  regulation.  A  transformer  to  regulate  well 
must  have  low  resistance  coils  and  little  magnetic  leakage.  Large 
transformers  as  a  rule  regulate  more  closely  than  small  ones. 

115.  Effect  of  resistance  of  coils  upon  the  action  of  a  trans- 
former.— Fig.  138  shows  the  general  effect  of  the  resistances  of 
the  coils  upon  the  action  of  a  transformer.  The  line  <9<E>  repre- 
sents the  harmonically  varying  flux  in  the  core.  Oa  represents 
the  useful  part  of  the  primary  electromotive  force  and  Ob  the 
total  electromotive  force  induced  in  the  secondary  coil.  The 
line  01"  represents  the  secondary  current  and  the  line  01' 

*  The  part,  namely,  which  balances  the  electromotive  force  induced  in  the  primary 
coil  by  the  magnetic  flux  which  passes  through  both  coils. 

f  The  lost  portions  of  primary  and  secondary  electromotive  forces  are,  in  general, 
not  in  phase  with  total  primary  and  total  secondary  electromotive  forces.  These 
losses  are  therefore  to  be  subtracted  as  vectors  as  explained  in  the  following  articles. 


174 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


represents  the  primary  current.  The  total  primary  electromotive 
force  E1  exceeds  Oa  by  the  amount  R'  I'  (parallel  to  /'),  and  the 
electromotive  force  E"  at  the  terminals  of  the  secondary  coil  falls 
short  of  Ob  by  the  amount  R"I"  (parallel  to  /"). 

Remark.  —  When  the  angle  6,  Fig.  138,  is  nearly  zero  (second- 
ary receiving  circuit  non-inductive)  then  R'  I'  and  R"I"  are 

nearly  parallel  to  Oa  and  Ob  re- 
spectively, so  that  Oa  is  much  less 
than  E'  in  value  and  E"  is  much 
less  than  Ob  in  value.  On  the  other 
hand,  when  the  angle  0  is  nearly 
db  90°  (secondary  receiving  circuit 
containing  a  large  inductance  or  a 
condenser)  then  Rf  P  and  R"  I" 
are  nearly  perpendicular  to  Oa  and 
Ob  respectively,  so  that  Oa  is  nearly 
equal  to  E'  in  value  and  E"  is 
nearly  equal  to  Ob  in  value. 
Therefore  the  regulation  of  a 
transformer  is  largely  affected  by 
coil  resistance  when  the  secondary  receiving  circuit  is  non-inductive, 
but  scarcely  at  all  affected  by  the  coil  resistance  when  the  second- 
ary receiving  circuit  contains  a  large  inductance  or  a  condenser. 

116.  Effect  of  magnetic  leakage  upon  the  action  of  a  trans- 
former. —  It  is  shown  in  the  next  article  that  magnetic  leakage  is 
in  its  effects  equivalent  to  an  auxiliary  outside  inductance  P 
through  which  the  primary  current  passes  on  its  way  to  the  pri- 
mary of  the  transformer.  The  part  of  the  primary  electromotive 
force  E'  which  is  lost  in  this  inductance  is  equal  to  <oPff  and  it 
is  90°  ahead  of  the  primary  current  /'  in  phase. 

When  the  secondary  receiving  circuit  is  inductive  Pf  lags  be- 
hind E"  (Ob,  Fig.  139)  by  the  angle  6  as  shown  in  Fig.  139,  and 
the  useful  part  Oa  of  the  primary  electromotive  force  is  less  than 
the  total  primary  electromotive  force.  In  this  case  the  secondary 


THE   TRANSFORMER. 


175 


1\ 

electromotive  force,  which  is  equal  to  -^  x  Oa,  falls  off  in  value 

as  I'  (and  also  coPP)  increases. 

When  the  secondary  receiving  circuit  contains  a  condenser,  I" 
is  ahead  of  E"  (Ob,  Fig.  140)  as  shown  in  Fig.  140,  and  the  use- 


Fig.  139. 


Fig.  140. 


ful  part,  Oa,  of  the  primary  electromotive  force  is  greater  than 
the  total  primary  electromotive  force  in  value.      In  this  case  the 

N" 

secondary  electromotive  force,  which  is  equal  to  -^7  x   Oa,  in- 
creases in  value  as  /'  (and  also  a)Pf)  increases. 

When  the  secondary  receiving  circuit  is  non-inductive  the 
angle  6  is  zero  and  <oPI'  is  at  right  angles  to  Oa,  so  that  Oa  is 
sensibly  equal  to  Ef  in  value,  and  therefore  sensibly  constant. 
In  this  case  the  secondary  electromotive  force  remains  sensibly 
constant  as  I'  (and  also  as  toPIf)  increases. 

117.  Proposition. — The  effect  of  magnetic  leakage  in  a  trans- 
former is  equivalent  to  a  certain  outside  inductance  P,  connected  in 
series  with  the  primary  coil. 

Discussion. — Let  A,  Fig.  141,  be  the  primary  coil,  B  the  sec- 
ondary coil  and  C  the  iron  core  of  a  transformer.  As  the  (har- 
monic) alternating  currents  in  A  and  B  pulsate,  harmonically 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


varying  fluxes  are  produced  through  the  core  and  around  the 
coils.  Let  OC,  Fig.  142,  represent  the  harmonically  vary- 
ing flux  through  the  core, 
Op  the  harmonically  vary- 
ing flux  which  encircles  coil 
A  only,  and  Os  the  har- 
monically varying  flux 
which  encircles  coil  B  only. 
The  fluxes  Op  and  Os  are 
proportional  to  and  in  phase 
with/'  and  /"respectively, 


so  that  the  total  flux  Op  -f 
Os  (represented  by  the  lines 
sp  or  bat  Fig.  142)  which 

passes  between  A  and  B  is  proportional  to  and  in  phase  with  /'.* 
The  total  harmonically  varying  flux  through  coil  A  is  OC  + 
Op  [=  Oa~\ ,  and  the  total  harmonically  varying  flux  through  coil 
B  is  OC+  Os  [=  Ob~\ .  Now,  Oa  =  Ob  -f  ba,  so  that  we  may  look 
upon  the  action  of  the  trans- 
former as  due  to  the  flux  Ob  pass- 
ing through  both  coils  and  the  flux 
ba  passing  through  the  primary 
coil  only.  This  latter  flux  being 
proportional  to  the  primary  cur- 
rent is  equivalent  in  its  effects  to 
an  inductance  P,  connected  in 
series  with  the  primary  coil.  Let 

<£'  be  the  value  of  the  leakage  flux  ab,  which,  for  a  given  value 
if  of  the  primary  current,  encircles  the  primary  coil,  then,  accord- 
ing to  equations  (5)  and  (6),  we  have 


118.  The  constant  current  transformer. — A  transformer  of  which 

*  Since  I'  is  proportional  to  I"  and  opposite  to  it  in  phase. 


THE   TRANSFORMER. 


177 


the  leakage  inductance  P  is  very  large  is  sometimes  called  a  con- 
stant current  transformer  for  the  reason  that  the  current  delivered 
by  such  a  transformer  varies  but  little  with  the  resistance  of  the 
receiving  circuit,  so  long  as  this  resistance  is  comparatively  small, 
the  primary  of  the  transformer  being  connected  to  constant  elec- 
tromotive force  mains.  The  action  of  the  inductance  P  in  con- 
trolling the  current  is  explained  in  the  article  on  the  constant 
current  alternator.  (Article  83.) 

Fig.  143  is  a  sketch  of  the  General 
Electric  Company's  type  of  constant 
current  transformer ;  C  is  the  iron 
core,  PP  the  primary  coil,  and  55 
the  secondary  coil.  The  secondary 
coil  is  movable  and  nearly  counter- 
balanced, and  the  increased  repulsion 

between  PP  and  55  due  to  a  slight  increase  of  current  lifts  the 
secondary  coil  to  5 '5'. 

When  the  primary  and  secondary  coils  are  near  together  the 
leakage  inductance  is  very  small  and  a  decrease  in  the  resistance 
of  the  receiving  circuit  would  be  accompanied  by  a  great  increase 
of  current  were  it  not  for  the  movement  of  the  secondary  coil  and 
the  consequent  increase  of  leakage  inductance. 

119.  Calculation  of  leakage  inductance  P. — The  leakage  flux  <£>' 
equation  (73)  [  =  ba,  Fig.  142],  and  therefore  the  value  of  P 
also  depends  upon  the  size  and  shape  of  the  primary  and  sec- 
ondary coils  and  upon  their  proximity  to  the  core.  In  consider- 
ing the  flux  between  the  coils  (leakage  flux)  we  need  not  consider 
whether  a  given  portion  of  this  flux  is  a  part  of  Op  or  a  part  of 
Os,  Fig.  142,  inasmuch  as  these  two  fluxes  are  added  together  to 
give  ba  or  <£>'. 

Figs.  144  and  145  show  side  and  end  views  of  a  shell  type 
transformer.  The  trend  of  the  leakage  flux  is  shown  in  the  upper 
part  of  Fig.  145  (omitted  from  lower  part  for  the  sake  of  clear- 
ness), and  the  dimensions  Xt  Y,  g,  \  and  /  are  shown.  Fig.  146 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


is  an  enlargement  of  the  upper  part  of  Fig.  145.      Consider  the 
flux  across  between  the  dotted  lines  aa,  Fig.  146.     The  magnet- 


Fig.    144. 

X 

omotive  force  pushing  this  flux  across  is  4717  '  -v^^V'.*    The  length 

J\. 

of  the  air  portion  of  the  magnetic  circuit  through  which  the  leak- 

age flux  flows  is  /  and  its  sectional 
area  is  \dx  (counting  both  limbs  of 
the  coils).  Therefore,  the  magnetic 
reluctance  of  this  leakage  circuit  is 


r-r- 

\dx 


and  the  flux  across  between  aa  is 


m.mf.  \xdx 

ff 


m.r. 


Fig.  145. 


This    flux    encircles   the   fractional 
part  -y.  of  the  primary  turns  and,  there- 


fore,  the  fractional  part  -^  of  the  flux  is  to  be  counted  as  encirc- 
J\. 

ling  the  entire  primary  coil  so  that 


*A11  quantities  in  this  article  are  expressed  in  c.g.s.  units. 


THE   TRANSFORMER. 


179 


lea fas&  flu,/ 


iron 


r    Iron 


primary 
a 

Jbf  ', 
'  Secondary 

\\ 
}l 

iron                i 

»x     iron 

;' 

Fig.  146. 


The  part  of  4>  '  which  flows  across  the  primary  coil  is  the  in- 
tegral of  this  expression  from  x  =  o  to  x  =  X.  This  part  of  <£  ' 
is  therefore 


Similarly,  the  part  of  <£>'  which  flows  across  the  secondary  coil  is 


The  flux  across  the  gap  g  between  the  primary  and  secondary 
coils  is  all  counted  as  a  part  of  <!>',  and  is  equal  to 


Therefore 


4irN'?\ 


Y 


(74) 


There  is  some  leakage  flux  passing  between  the  primary  and 
secondary  coils  where  they  project  beyond  the  iron  core.  This 
part  of  the  leakage  flux  has  a  longer  air-path  than  the  leakage 
flux  which  flows  from  iron  to  iron,  say  three  times  as  long. 
Therefore,  for  X  we  may  take  the  total  length  of  the  coils  less- 
ened, by  say  2/^  the  length  which  is  surrounded  by  air  only. 

Substituting  the  value  of  <£>'  from  (74)  in  (73),  we  have 

"Prf+f+^l        (75) 


P: 


In  this  expression  N'if  being  equal  to  N"i"  is  written  therefor. 


i8o 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


This  equation  gives  the  value  of  P  is  centimeters,  all  dimen- 
sions being  expressed  in  centimeters. 

The  equivalent  inductance  P  may  be  reduced  in  value  by  less- 
ening X,  X,  Y  or  g  or  by  increasing  /.  Fig.  147  shows  the  pro- 
portions of  a  recent  type  of 
transformer  for  which  the 
leakage  inductance  P  is  very 
small.  The  value  of  P  may 


X  T 


Fig.  147. 


be  further  reduced  by  wind- 
ing the  primary  and  sec- 
ondary coils  in  alternate 
sections. 

120.  Formulation  of  the 
complex  equations  of  the 
transformer. — The  following 
discussion  is  taken  from 
Steinmetz,*  with  the  follow- 
ing changes.  A  circuit  which 
takes  a  lagging  current  has 
its  impedance  written  r  +jx 
and  its  admittance  written 
g  —  jb.  Magnetic  leakage 
is  here  represented  as  equiv- 
alent to  a  primary  induct- 
ance only,  while  Steinmetz 
uses  both  primary  and  sec- 


ondary inductance.     The  notation  is  slightly  altered. 
Let  Nf  =  number  of  primary  turns. 
N"  =  number  of  secondary  turns. 
N^ 

=  W 

rl  =  resistance  of  primary  coil. 
r2  =  resistance  of  secondary  coil. 


*  "Alternating  Current  Phenomena,"  third  edition,  page  204. 


THE   TRANSFORMER.  l8l 

=  reactance  value,  a>P,  of  the  primary  leakage  induct- 
ance P. 


Y=g  —  jb  —  admittance  of  entire  secondary  circuit  includ- 
ing resistance  of  secondary  coil. 

Ef  —  primary  impressed  electromotive  force. 

A  =  that  part  of  E'  which  is  used  to  balance  the  electromo- 
tive force  induced  in  the  primary  coil  by  the  mag- 
netic flux  which  passes  through  both  coils. 

B  =  secondary  induced  electromotive  force. 
E"  =  secondary  terminal  voltage. 

/'  =  total  primary  current. 

M  =  magnetizing  current. 

/"  =  secondary  current. 

M 

Yl  ==  gl  —jb^  =  —r  =  admittance  of  transformer  at  zero  out- 

put. 

M 
Remark.  —  In  Article   112,  Yl  is  defined  as  -p-r    The  mutual 

influences  of  coil  resistances,  eddy  currents  and  hysteresis,  and 

M  . 
magnetic  leakage  are  partly  taken  account  of  by  using  —j  instead 

M 
of  ^,  because  the  extent  of  magnetization  of  the  core  depends 

upon  A. 

The  ratio  of  A  and  B  is  a  and  they  are  opposite  in  phase,  so  that 

A  =  -  aB  (i) 

The  secondary  current  is 

I"  =  YB  (ii) 

The  part  of  the  primary  current  corresponding  to  I"  is  opposite 

YB 

to  it  in  phase  and  I/a  as  large.     It  is  therefore  equal  to  --  —  , 

which  added  to  the  magnetizing  current  M  (=  Y^A),  gives 

YB 

r  =  YIA  -  - 


1 82  ELEMENTS   OF   ALTERNATING   CURRENTS. 

The  electromotive  force  lost  in  the  secondary  coil  is  rj",  so 
that  the  secondary  terminal  voltage  is 

E"  =  B-  rj"  (iv) 

The  electromotive  force  used  to  overcome  the  impedance  Z  of 
the  primary  coil  is  Z/',  which  added  to  A  gives  the  primary  im- 
pressed electromotive  force.  That  is, 

E'  =  A  +  ZP  (v) 

With  the  help  of  equations  (i)  and  (ii),  I",  /',  E"  and  E1 ',  may 
be  expressed  in  terms  of  B,  giving 

(76) 

(77) 
'  (78) 

(79) 

It  is  somewhat  more  desirable  to  express  /",  /'  and  En ',  in 
terms  of  E' .  Eliminating  B  from  equations  (78)  and  (79),  we  have 

E"  l—r2Y 


7V 
a  +  aZY,  +  - 


or 


Similarly  we  find 


and 


a    ___  pi 


a2  +  a2ZFx  +  ZF 


For  purposes  of  numerical  calculation  E'  may  be  taken  as  the  reference  axis,  so 
that  Ef  becomes  a  simple  quantity  and  it  remains  only  to  separate  the  components  of 
the  various  factors  by  which  E'  is  multiplied.  Take  for  example  the  expression  for 


THE   TRANSFORMER.  183 


I".     Using  g  —  jo  for  Y,  rv  -\-j'jc  for  Z,  and^j  —  jbl  for  Yv  and  collecting  terms  we 
have 

factor  in  equation  (  82)  =  —  ^     Ja 
u+jv 

where 

«  =  a2  +  a2^^  -(-  ^TJ  +  a2^*  +  bx 

v  =  a^-jjc  -f  gx  —  a^jfj  —  £rj 

Multiplying  numerator  and  denominator  by  u  —  jv  we  have 

agu  —  abv  . 


actor  =- 

So  that  we  have  the  mimerical  relation 

numerical  value  of  I"  =  Vs*^-fl  -  E' 
where 

agu  —  abv 


abu-\-  agv 

: 


The  numerical  values  of  7X  and  I"  may  be  calculated  in  a  similar  manner  from  the 
known  values  of  a,  rv  x,  g^  bv  and  rv  and  given  values  of  E't  g,  and  b.  If  r3  and 
x3  are  the  resistance  and  reactance  respectively  of  the  external  secondary  circuit,  then 


*     . 
and 


Calculation  of  regulation..  —  The  falling  of  the  secondary  ter- 
minal voltage  of  a  well-designed  transformer  depends  almost 
wholly  upon  magnetic  leakage  and  coil  resistances,  and  is  to  a 
very  close  degree  of  approximation  independent  of  the  magnetiz- 
ing current.  Therefore  for  the  purpose  of  the  calculation  of 
transformer  regulation  Yl  may  be  assumed  to  be  equal  to  zero. 
Under  these  conditions  equation  (80)  becomes 


from  which,  by  separating  the  components  of  the  factor 


a-ar2Y 


the  numerical  value  of  E"  may  be  calculated  for  any  assigned 
value  of  Y',  a,  rv  and  Z  being  known. 


1 84  ELEMENTS   OF   ALTERNATING   CURRENTS. 

PROBLEMS. 

93.  Following  are  the  data  for  a  shell  type  transformer  (see 
Figs.  144,  145  and  146): 

The  fine  wire  coil  consists  of  560  turns  of  number  17  B.  &  S. 
copper  wire.  Mean  length  of  turn  29^  inches. 

The  coarse  wire  coil  consists  of  28  turns  of  two  number 
sevens  (B.  &  S.).  Mean  length  of  turn  2gl/2  inches. 

/=  i  y^  inches,  g  —  y&  inch,  Y—  i  J^  inches  (coarse  wire  coil), 
X=  i  y2  inches  (fine  wire  coil),  and  X/2  =  10^  inches. 

At  each  end  of  core  4}^  inches  of  length  of  coils  are  exposed. 
Section  of  magnetic  circuit  =  10^2  x  iy|  inches  and  -f$  of  this 
is  iron.  Volume  of  iron  =24^  square  inches  x  ioy£  inches 
X  i8Q-.  Thickness  of  laminations  14  mils. 

When  the  fine  wire  coil  takes  current  at  1,100  volts  at  133 
cycles  per  second  find  the  secondary  terminal  voltage  at  zero  load. 

Find  the  secondary  terminal  voltage  when  the  transformer  is 
delivering  current  to  a  non-inductive  receiving  circuit  of  1.8 
ohms  resistance. 

Find  the  secondary  terminal  voltage  when  the  transformer  is 
delivering  current  to  a  circuit  of  which  the  resistance  is  1.25 
ohms  and  the  reactance  is  1.3  ohms. 

Find  the  secondary  terminal  voltage  when  the  transformer  is 
delivering  current  to  a  resistanceless  circuit  of  which  the  react- 
ance is  1.8  ohms. 

94.  Following  are  the  data  for  a  core  type  transformer  (see 
Fig.  147) : 

The  fine  wire  coil  consists  of  2,000  turns  of  number  17  B.  &  S. 
copper  wire.  Length  of  mean  turn  =  9  y2  inches.  This  coil  is 
wound  next  the  core. 

The  coarse  wire  coil  consists  of  100  turns  of  two  number 
sevens  (B.  &  S.).  Length  of  mean  turn  1 1  ^  inches.  This  coil 
is  wound  outside  of  the  fine  wire  coil. 

//2  =  5  inches,  g—%  inch,  F=  ^  inch  (coarse  wire  coil), 
X  =  y±  inch  (fine  wire  coil),  and  X=  11^  inches. 


THE   TRANSFORMER.  185 

The  net  sectional  area  of  the  iron  core  is  3.7  square  inches, 
the  mean  length  of  the  magnetic  circuit  is  22  inches,  and  the 
volume  of  the  iron  is  81.4  cubic  inches.  Thickness  of  lamina- 
tions 14  mils. 

When  the  fine  wire  coil  takes  current  at  1,100  volts  at  133 
cycles  per  second  find  the  secondary  terminal  voltage  at  zero  load. 

Find  the  secondary  terminal  voltage  when  the  transformer  is 
delivering  current  to  a  non-inductive  receiving  circuit  of  which 
the  resistance  is  1.8  ohms. 

Find  the  secondary  terminal  voltage  when  the  transformer  is 
delivering  current  to  a  circuit  of  which  the  resistance  is  1.25 
ohms  and  the  reactance*  is  1.3  ohms. 

Find  the  secondary  terminal  voltage  when  the  transformer  is 
delivering  current  to  a  resistanceless  circuit  of  which  the  react- 
ance is  1.8  ohms. 


CHAPTER   XII. 


noo 


THE  SYNCHRONOUS  MOTOR. 

121.  Alternators  in  series. — Two  alternators  A  and  B  are  con- 
nected in  series  and  driven  by  separate  engines  to  give  precisely 
the  same  frequency;  The  lines  A  and  B,  Fig. 
148,  represent  the  electromotive  forces  of 
machines  A  and  B  respectively,  <f>  is  the  an- 
gular lag  of  the  electromotive  force  B  behind 
the  electromotive  force  A,  and  the  line  E  rep- 
resents the  resultant  electromotive  force  of  A 
and  B.  This  resultant  electromotive  force 
produces  in  the  circuit  a  current,  of  which  the 
value  is 

i          (83) 


and  which  lags  0°  behind  E  in  phase,  where 

coL 
tan  0  =  -=  (84) 

Fig.  148. 

in  which  R  is  the  resistance  of  the  circuit,  L  is  the  total  induc- 
tance of  the  circuit  including  the  armatures  of  both  machines, 
and  a)  (=  27T/")  is  the  frequency  in  radians  per  second. 
The  power  P'  put  into  the  circuit  by  machine  A  is 

where  (AI)  is  the  angle  between  A  and  7.     The  power  put  into 
the  circuit  by  machine  B  is 

P"  =  BIcos(BI)  (86) 

1 86 


THE   SYNCHRONOUS    MOTOR. 


I87 


The  angle  AI  in  Fig.  148  is  less  than  90°,  so  that  cos  (AI)  is 
positive ;  therefore  P'  is  positive,  that  is,  the  machine  A  is  acting 
as  a  dynamo.  The  angle  (BI)  in  the  figure  is  greater  than  90°, 
so  that  cos  (BI)  is  negative ;  therefore  P"  is  negative,  that  is,  the 
machine  B  is  acting  as  a  motor. 

The  alternator  B  used  in  this  way  is  called  a  synchronous  motor, 
the  alternator  A  being  driven  by  an  engine  or  water-wheel. 

122.  Variation  of  P'  and  P"  with  the  phase  angle  <£. — Draw  a 
line  OC,  Fig.  149,  representing  B  to  scale.  Describe  about  C  a 


Fig.  149. 

circle  of  which  the  radius  represents  A.  Then  a  line  OP,  from 
O  to  any  point  in  the  circle  represents  a  possible  value  of  the  re- 
sultant electromotive  force  E,  <f>  being  the  corresponding  phase 
difference  between  A  and  B.  Draw  the  line  ef  through  0,  mak- 
ing with  O  C  the  angle  0.  Then  the  angle  POfis  equal  to  the 
angle  (BI).  Therefore 

OQ=OP  cos  (BI)  =  E  cos  (BI)  =  */R2  +  o>2Z2  /  cos  (BI) 


188 
or 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


co2L2 


Substituting  this  value  of  /  cos  (BI)  in  equation  (86)  we  have 


That  is,  the  output  of  the  machine  B  is  proportional  to  the 
projection  OQ  of  the  line  OP  on  the  line  ef,  B  and  </&  +  to2!2 
being  constant. 

When  Q  is  towards  /  from  0,  cos  (BI)  is  positive  so  that  P" 
is  positive  and  machine  B  acts  as  a  dynamo.  When  Q  is  towards 
e  from  O,  cos  (BI)  is  negative  so  that  P"  is  negative  and  machine 
B  acts  as  a  motor. 


Fig.  150  is  a  construction,  for  the  same  value  of  <£,  in  which 
Of  Q'  represents  the  power  P'  put  into  the  circuit  by  machine  A. 
From  this  diagram  we  have 


A-O'Q' 


(88) 


THE   SYNCHRONOUS   MOTOR.  189 

The  projection  of  A,  Fig.  149,  on  ef  is  A  cos  (<£  —  6)  ;  the  pro- 
jection of  B  on  efis  B  cos  0  ;  and  OQ  is  the  sum  of  these  pro- 
jections so  that 

OQ  =  A  cos  (<f>  —  6)  +  B  cos  0 


Substituting  this  value  of  OQ  in  equation  (87)  we  have 
AB  B1 


(89) 


Similarly  from  Fig.  150  we  have 

AB  A2 


pr  - 


These  equations,  (89)  and  (90),  are  the  fundamental  equations 
of  the  synchronous  motor.  The  algebraic  sum  of  the  outputs  of 
machines  A  and  B  is,  of  course,  equal  to  RI2,  so  that 

p'+p>'  =  Rr  (9i) 

This  relation  may  be  derived  from  equations  (89)  and  (90),  re- 
membering that 

a>L  R 

sin  6  =     ,-^— ==      cos  0  = 


and 

E2  A2  +  B*  +  2AB  cos 


_ 

" 


The  ordinates  of  the  curves  P',  P"'and  RI2,  Fig.  151,  show 
the  values  of  P1  ',  of  P"  and  of  .AY2  for  values  of  <f>  from  zero  to 
360°.  Positive  ordinates  represent  positive  power  (dynamo  ac- 
tion), negative  ordinates  represent  negative  power  (motor  action). 
Each  ordinate  of  the  curve  RI2  is  the  algebraic  sum  of  the  ordi- 
nates of  the  curves  P'  and  P"  ;  Fig.  152  shows  portions  of  the 
curves  P'  ,  Pn  and  RI2  to  a  larger  scale.  The  ordinates  of  the 

I  P"\ 
curve  77  represent  the  efficiency  I  ^7  j  of  transmissions  for  vari- 

ous values  of  <f>  when  machine  B  is  a  motor. 


190  ELEMENTS   OF   ALTERNATING   CURRENTS. 


3000, 

^- 

~^x 

/^~ 

^N 

/ 

s\ 

/I' 

1000 

\ 

/ 

^  — 
iSoo 

—  -^ 

XT 

~c. 

\ 

/  / 

-T 

—  -^ 

>£- 

/ 

X 

N 

X 

^ 

S\ 

// 

/ 

^ 

5oo 

\ 

^ 

\ 

// 

/ 

/ 

0 

X 

s 

s^ 

^ 

/ 

/ 

/ 

-500 

r 

^  

-~r 

0          JO           *0          90         l'4 

0        /50         180         2. 

0       21 

o       3,*} 

a     jao      jj 

0       Ji 

0 

7»jTa««  Difference 

A  =  1,100  volts.  J?=  i.oo  ohm. 

B  =  1,000  volts.          wZ  —  0.58  ohm. 

Fig.  151. 


100 


\ 


80% 


60 


ZOO 


OO 


zo 


-100 


/go  ^l/w  *f  Q       —  y) 

^4  =  I,ioo  volts.  R  =  I.oo  ohm. 

B  =  I ,  loo  volts.          w  Z  =  o.  58  ohm. 

Fig.   152. 

Steinmetz's  derivation  of  equations  (89)  and  (90). — Let  B  be  the  electromotive 
force  of  machine  By  and  let  this  electromotive  force  be  taken  as  the  .r-axis  of  refer- 


THE   SYNCHRONOUS    MOTOR.  I91 

ence.  Let  fy  be  the  angle  that  the  electromotive  force  of  machine  A  is  ahead  of  B, 
and  let  A  be  the  numerical  value  of  the  electromotive  force  of  machine  A.  Then 
A  cos  <j>  is  the  x- component  of  the  electromotive  force  of  machine  A,  and  A  sin  <j>  is 
its  ^-component.  Therefore  the  complex  expression  for  the  electromotive  force  of 
machine  A  is  A  cos  0  -\-jA  sin  0,  and  the  resultant  electromotive  force  of  the  two 

machines  is : 

£  =  B  -f  A  cos  ty  -\-jA  sin  <j> 

The  current  is  therefore 

E  B  -\-  A  cos  0  4-  j'A  sin  <j> 


R+juL  R  -\-juL 

The  real  part  of  /  is  the  component  of  /  parallel  to  B,  and  the  product  of  this 
component  of  7  into  B  is  the  power  output  of  machine  B,  therefore 


_  B*R  +  ABR  cos 


123.  The  necessity  of  synchronism  for  the  operation  of  machine 
B  as  a  motor.  —  In  Fig.  148  the  relative  phase  of  the  electromo- 
tive forces  A  and  B  has  been  so  chosen  that  machine  B  is  a 
motor.    We  shall  now  consider  whether  synchronism  of  machines 
A  and  B  is  a  necessary  condition  for  the  steady  *  intake  of  power 
by  the  machine  B.     Suppose  that  machine  B  is  running  steadily 
(engine  driven)  at  a  frequency  slightly  above  or  below  the  fre- 
quency of  A.     Then  the  phase  angle  </>  will  change  continuously 
and  the  point  Pt  Fig.  153,  will  move  slowly  around  the  circle, 
making  one  revolution  while  machine  B  gains  or  loses  one  cycle 
with  reference  to  A,  and  the  power  intake  P"  of  machine  B  will 
pass  through  a  set  of  positive  and  negative  values  during  each 
revolution  of  the  point  P.     Now  the  positive  values  of  P"  are 
larger  than  the  negative  values,  and  also  of  longer  duration,  so 
that  the  average  value  of  P"  is  positive.     That  is,  machine  B  gives 
out  power  on  the  average  if  it  is  either  above  or  below  synchro- 
nism with  A.     Therefore  synchronism  is  a  necessary  condition 
for  the  intake  of  power  by  machine  B. 

124.  Behavior  of  a  synchronous  motor.     Stoppage  due  to  over- 
load. —  When  machine  B  is  running  as  a  motor,  unloaded,  its 

*  That  is,  steady  except  for  the  extremely  rapid  pulsations  due  to  the  alternations 
of  electromotive  force  and  current. 


192 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


intake  P"  is  approximately  zero  ;  point  P,  Fig.  153,  is  at  £*; 
the  resultant  electromotive  force  is  Os ;  the  current  is  in  quadra- 
ture with  the  electromotive  force  B ;  and  the  output  of  machine 
A  is  equal  to  RP. 

As  the  motor  B  is  loaded,  its  intake  P  "  increases  ;  the  point 
Py  Fig.  153,  moves  from  s  towards  M;  and  the  resultant  electro- 


Fig.  153. 

motive  force,  the  current,  and  RI2,  all  grow  less  until  <£=  180°. 
Further  loading  of  B  carries  the  point  P  further  towards  M,  and 
the  resultant  electromotive  force,  the  current,  and  RI2,  all  in- 
crease. When  the  point  P,  Fig.  153,  reaches  the  line  ef,  the 
current  is  opposite  to  the  electromotive  force  B  in  phase.  As  the 
motor  B  is  still  further  loaded  the  point  P  moves  on  towards  M, 
and  when  the  point  P  reaches  M  the  intake  of  B  has  reached 

*  Running  of  B  is  unstable  when  the  point  P  is  at  s'. 


THE   SYNCHRONOUS   MOTOR  193 

its  maximum  value  for  the  given  values  of  A,  B,  a>L  and  <R. 
Further  loading  of  B  decreases  its  intake.  This  decrease  of  intake 
causes  B  to  fall  further  and  further  behind  A  in  phase  until  the 
point  P  moves  beyond  the  point  s'  ;  machine  B  then  acts  as  a 
dynamo  and  gives  out  power  to  the  line,  which  action,  together 
with  the  belt  load,  causes  machine  B  to  fall  out  of  synchronism 
and  stop.  While  machine  B  is  stopping  the  action  is  as  follows  : 
Every  time  machine  B  loses  one  cycle  as  compared  with  the 
steadily  driven  machine  A,  the  point  P,  Fig.  153,  moves  once 
around  the  circle.  While  P  is  moving  from  sf  through  D  to  s 
machine  B  acts  as  a  dynamo,  which  action,  together  with  the 
belt  load,  slows  the  machine  up  greatly.  Then  as  P  moves  from 
s  through  M  to  s' ,  machine  B  takes  in  power  from  A,  but  by  no 
means  enough  to  enable  it  to  regain  synchronous  speed.  There- 
fore the  point  P  moves  on  past  s'  through  D  to  s,  which  slows 
up  machine  B  still  more,  and  so  on. 

125.  Behavior  of  a  synchronous  motor  of  which  the  electro- 
motive force  is  greater  than  the  electromotive  force  of  the  gen- 
erator which  drives  it. — In  this  article  we  shall  speak  of  machine 
A  as  the  motor  and  of  machine  B  as  the  generator,  machine  A 
having  the  greater  electromotive  force.  It  is  evident  from  Figs. 

151  and   152  that  machine  A  can  act  as  a  motor.     From  Fig. 

152  it  is  also  evident  that  the  maximum  intake  P'  of  A  as  a 
motor  is  very  much  less  than  the  maximum  intake  of  B  as  a 
motor  and  it  is  also  evident  that  when  A  is  acting  as  a  motor  its 
intake  P'  is  much  less  than  the  output  P"  of  B,  so  that  the  effi- 
ciency of  transmission  is  lower  when  A  is  the  motor  than  when 
B  is  the  motor.     When  machine  A  is  running  as  a  motor,  un- 
loaded, its  intake  Pf  is  approximately  zero  ;  the  point  P,  Fig. 
1 54,  is  at  s ;  the  resultant  electromotive  force  is  Os ;  the  current 
is  in  quadrature  with  the  electromotive  force  A ;  and  the  output 
of  the  generator  B  is  equal  to  RI2. 

As  the  motor  A  is  loaded,  its  intake  P'  increases ;  the  point 
P,  Fig.  1 54,  moves  from  s  towards  M\  and  the  resultant  electro- 


I94 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


motive  force,  the  current,  and  RP  all  increase.     When  the  point 
P  reaches  M  the  intake  of  A  has  reached  its  maximum  value  for 


Fig.  154. 

the  given  values  of  A,  B,  wL  and  R,  and  further  loading  causes 
machine  A  to  fall  out  of  synchronism  and  stop. 

Negative  reactance  of  machine  A. — Any  receiving  circuit  which 
takes  current  which  is  ahead  in  phase  of  the  electromotive  force 
of  the  generator  which  supplies  the  current  acts  more  or  less  like 
a  condenser  and  has  negative  reactance.  A  careful  scrutiny  of 
Fig.  1 54  will  show  that  /  is  ahead  of  B  in  phase  when  the  point 
P  is  near  s. 

126.  The  starting  of  the  synchronous  motor. — Let  A  be  the 
machine  which  is  to  be  the  generator  and  B  the  machine  which 
is  to  be  the  motor.  The  machine  A,  which  is-  driven  continu- 
ously by  an  engine  or  water-wheel,  is  started.  Machine  B  is  then 
started  by  driving  it  with  an  engine  or  other  independent  mover, 
and  its  speed  is  carefully  regulated  until  (a)  it  is  in  synchronism 
with  A,  and  (b)  its  electromotive  force  opposite  to  the  electromotive 


THE   SYNCHRONOUS    MOTOR.  195 

force  of  A  in  phase.  The  circuit  is  then  closed  and  the  engine  or 
other  agent  which  has  been  used  to  bring  machine  B  up  to  speed 
may  be  disconnected. 

In  carrying  out  this  process  it  is  necessary  to  use  some  sort  of 
indicator  for  showing  when  A  and  B  are  in  synchronism  and  when 
they  are  in  the  proper  phase  relation.  For  example,  a  lamp  may 
be  connected  in  circuit,  as  shown  in  Fig.  155.  This  lamp  pul- 


Fig.   155. 

sates  in  brightness  as  machine  B  is  being  speeded  up  and  the 
pulsations  become  slower  and  slower  as  the  frequency  of  B  ap- 
proaches the  frequency  of  A.  When  the  lamp  is  at  its  maximum 
brightness  the  angle  <£,  Fig.  149,  is  zero,  and  when  the  lamp  is  at 
its  minimum  of  brightness  the  angle  </>  is  180°.  When  the  pul- 
sations become  very  slow  the  machines  are  practically  in  syn- 
chronism and  the  switch  s  is  closed  when  the  lamp  is  at  minimum 
brightness. 

In  practice  the  lamp  /  is  connected  in  series  with  the  secon- 
daries of  two  transformers  the  primaries  of  which  are  connected 
to  A  and  to  B  respectively.  With  this  arrangement  the  lamp 
may  be  either  at  its  maximum  or  at  its  minimum  of  brightness 
when  the  machines  are  in  the  proper  phase  for  the  closing  of  the 
switch,  according  to  the  connections  of  the  transformers.  In  fact 
it  is  better  to  arrange  the  connections  so  that  the  lamp  shall  be 
at  its  maximum  of  brightness  when  the  switch  is  to  be  closed, 
for  the  instant  of  maximum  brightness  is  more  sharply  marked 
than  the  instant  of  minimum  brightness. 

127.  Stability  of  running-  of  synchronous  motor. — Suppose  the 


196  ELEMENTS    OF   ALTERNATING   CURRENTS. 

machine  A  to  be  driven  by  means  of  a  governed  engine  at  con- 
stant speed,  irrespective  of  its  output ;  and  suppose  the  motor  B 
to  be  running  steadily.  If  the  load  on  B  is  suddenly  increased 
this  machine  will  run  momentarily  slower  than  A  and  fall  behind 
A  in  phase.  If  this  falling  behind  in  phase  increases  the  power 
which  B  takes  from  A,  then  B  will  fall  only  so  far  behind  as  to 
enable  it  to  take  in  power  sufficient  to  carry  its  increased  load. 
If,  on  the  other  hand,  this  falling  behind  in  phase  decreases  the 
power  which  B  takes  from  A,  then  B  will  fall  further  and  further 
behind  A,  fall  out  of  synchronism  and  stop.  In  the  first  case 
the  running  of  B  is  stable,  in  the  second  case  it  is  unstable. 

As  the  point  P,  Fig.  149,  moves  along  the  circle  in  the  direc- 
tion of  the  arrow  &>  the  electromotive  force  A  is  getting  farther 
ahead  of  B  in  phase  or  B  is  falling  behind  A.  If  the  motor  intake 
of  B  increases  as  it  falls  behind  B,  then  the  running  of  B  is  stable, 
and  vice  versa,  as  pointed  out  above.  Now  the  projection  of  OP, 
namely  0  Q  positive  towards  e,  represents  the  intake  of  B  ;  and 
this  intake  increases  from  s  to  M,  Fig.  153,  as  B  falls  behind  A, 
and  decreases  from  M  to  sf  as  B  falls  behind  A  ;  therefore  s  to  M 
is  the  region  of  stable  motor  running  of  B,  and  J/to  s'  is  the 
region  of  unstable  motor  running  of  B. 

If  B  is  running  with  given  load  as  a  motor  the  point  P  will 
take  up  a  position  between  s  and  M  such  that  the  intake  of  B  is 
sufficient  to  carry  its  load.  If  B  is  further  loaded  P  moves 
further  towards  M\  if  B  is  unloaded  P  moves  towards  s.  If  B 
is  loaded  until  P  reaches  M  then  further  loading  decreases  the  in- 
take of  B  and  the  machine  B  therefore  falls  out  of  synchronism 
and  stops,  as  has  been  explained. 

Remark. — With  given  motor  intake  the  point  P  may  be  in  the 
region  sM  of  stable  running  or  in  the  region  Ms'  of  unstable  run- 
ning. For  the  first  case  the  current  is  smaller  than  for  the  sec- 
ond inasmuch  as  the  resultant  electromotive  force  is  smaller  in 
the  first  case. 

128,  Running  of  two  alternators  in  parallel  as  generators. — 


THE   SYNCHRONOUS   MOTOR.  197 

Two  alternators,  both  engine-driven,  run  satisfactorily  as  genera- 
tors when  they  are  adjusted  to  synchronism  (and  to  proper  phase 
relation)  and  connected  in  parallel  to  receiving  mains,  as  shown 
in  Fig.  156.  This  arrangement  is  frequently  used  in  practice. 


main 


main 


Fig.  156. 

Machines  A  and  B  are  started  and  connected  together  through 
the  indicating  lamp  /.  The  machines  are  adjusted  to  synchro- 
nism, and  when  the  lamp  is  at  minimum  brightness  the  switch  s  is 
closed.  The  switch  s'  is  then  closed,  and  the  machines  deliver 
current  to  the  mains.  If  either  machine  should  fall  behind  the 
other  in  phase  its  share  of  the  load  is  reduced  so  that  the  run- 
ning of  the  two  machines  is  stable.  The  lamp  /  is  in  practice 
connected  in  series  with  the  secondaries  of  two  transformers,  the 
primaries  of  which  are  connected  to  A  and  to  B  respectively, 
and  the  connections  are  so  made  that  the  proper  conditions  for 
closing  the  switch  s  are  indicated  by  maximum  brightness. 

Machine  A  may  be  run  alone  when  the  output  of  the  station 
is  small,  during  the  day,  for  example ;  and  when  the  load  in- 
creases to  the  full  capacity  of  machine  A,  machine  B  may  be 
started  up,  adjusted  to  synchronism  with  A  and  connected  to  the 
mains  by  closing  the  switch  s,  the  switch  sf  being  left  closed 
during  the  whole  operation. 

Sharing  of  load  between  two  engine- driven  alternator  generators  in  parallel. — (a) 
When  one  engine  only  is  governed,  the  other  engine  being  set  at  a  fixed  cut-off.  In 
this  case  the  output  of  the  alternator  driven  by  the  fixed  cut-off  engine  is  constant  and 
the  variations  of  station  output  are  met  by  the  governed  engine.  If  the  station  out- 
put falls  below  the  constant  output  of  the  alternator,  which  is  driven  by  the  fixed  cut- 


198  ELEMENTS   OF   ALTERNATING   CURRENTS. 

off  engine,  the  other  alternator  takes  in  power  as  a  synchronous  motor,  it  may  even 
take  in  enough  power  to  drive  its  engine  and  if  the  station  output  falls  too  low  the 
fixed  cut-off  engine  may  cause  the  entire  system  to  race.  This  arrangement  is  seldom 
used  in  practice. 

(b)  When  both  engines  are  governed  the  distribution  of  load  between  them  is 
approximately  as  follows  :  Let  a  be  the  zero-load-speed  of  engine  A,  and  let  b  be  the 
zero-load-speed  of  engine  B.  Let  s  be  the  common  speed  of  both  engines  when 
they  are  driving  the  two  alternators.  Let  P'  be  the  power  delivered  by  engine  A  to 
its  alternator,  and  P"  the  power  delivered  by  engine  B  to  its  alternator.  Then, 
approximately  : 

P'=m(a-s)  (i) 

and 


These  equations  are  based  on  the  assumption  that  the  speed  of  a  governed  engine 
falls  off  in  proportion  to  its  output.  The  quantity  m  is  obtained  by  dividing  the  full 
load  output  of  engine  A  by  its  full  load  drop  in  speed.  The  quantity  n  is  found  by 
dividing  the  full  load  output  of  engine  B  by  its  full  load  drop  in  speed. 

The  total  station  output  determines  the  combined  output  P/-\-P//  of  the  two 
engines,  and  equations  (i)  and  (ii)  determine  P',  P"  and  s. 

The  engines  share  the  load  equally  only  when  their  zero  load  speeds  are  equal  and 
when  their  full  load  drop  in  speed  is  the  same. 

Remark.  —  Well  designed  engines  fall  off  but  very  little  in  speed  with  increase 
of  load. 

129.  Hunting  action  of  the  synchronous  motor.  —  When  the 
load  on  a  synchronous  motor  is  increased  the  motor  slows  up 
momentarily  and  falls  behind  the  generator  in  phase.  When 
the  motor  has  fallen  behind  sufficiently  to  take  in  power  enough 
to  enable  it  to  carry  its  load  it  is  still  running  slightly  below 
synchronism,  it  therefore  falls  still  further  behind  and  takes  an 
excess  of  power  from  the  generator  which  quickly  speeds  it  above 
synchronism.  It  then  gains  on  the  generator  in  phase  until  it 
takes  in  less  power  than  is  required  for  its  load  when  it  again 
slows  up  and  so  on.  This  oscillation  of  speed  above  and  below 
synchronism,  called  hunting  ^  is  similar  to  the  hunting  of  a  gov- 
erned steam  engine.  It  is  frequently  a  source  of  great  annoy- 
ance, especially  where  several  synchronous  motors  (or  rotary 
converters)  are  run  in  parallel  from  the  same  mains. 

Remark.  —  The  hunting  of  the  synchronous  motor  is  usually 
more  troublesome  in  case  of  the  rotary  converter  than  it  is  in 
case  of  the  synchronous  motor  with  a  belt  load.  In  Chapter 


THE   SYNCHRONOUS   MOTOR.  199 

XIII.  the  method  employed  for  damping  the  hunting  oscillations 
of  a  synchronous  motor  is  described. 

Theory  of  the  hunting  of  the  synchronous  motor.  —  When  a  synchronous  motor  is 
running  steadily  it  takes  in  power  steadily  from  the  mains  and  gives  out  power  steadily 
on  its  belt  (or  from  its  direct  current  commutator  in  case  of  the  rotary  converter). 
The  pulsations  of  power  intake  due  to  the  alternations  of  electromotive  force  and  cur- 
rent are  extremely  rapid  in  comparison  with  hunting  oscillations  and  need  not  be  con- 
sidered, indeed  these  pulsations  do  not  exist  in  case  of  polyphase  machines. 

The  mean  position,  at  a  given  instant,  of  the  armature  of  a  synchronous  motor 
which  is  hunting  is  the  position  it  would  have  at  that  instant  if  it  were  turning  at  a 
constant  angular  velocity.  When  the  motor  hunts  its  armature  oscillates  forwards 
and  backwards  through  its  mean  position. 

When  the  armature  is  in  its  mean  position  the  power  intake  of  the  motor  and  its 
belt  load  (including  friction  losses  and  the  like)  are  equal,  and  no  unbalanced  torque 
acts  on  the  armature. 

When  the  armature  gets  ahead  of  its  mean  position  its  intake  is  lessened,  the  belt 
load  of  the  machine,  which  is  assumed  to  be  constant,  exceeds  the  intake  and  an  un- 
balanced retarding  torque  acts  on  the  armature. 

When  the  armature  falls  behind  its  mean  position  its  intake  exceeds  its  belt  load 
and  an  unbalanced  accelerating  torque  acts  on  the  armature. 

Let  V'  be  the  angle  between  the  mean  position  of  the  armature  and  its  actual  posi- 
tion at  a  given  instant,  and  let  T  be  the  unbalanced  torque  acting  on  the  armature  at 
this  instant.  Our  problem  is  to  find  the  relation  between  ty  and  T.  This  relation, 
when  i/>  is  small  is 

T=  —  ty  (i) 

where  b  is  a  constant.     Therefore,  from  the  laws  of  harmonic  motion,  we  have 


in  which  K  is  the  moment  of  inertia  of  the  rotating  part  of  the  machine  and  /  is  the 
period  of  the  hunting  oscillations. 

Derivation  of  equation  (i).  —  We  shall  derive  equation  (i)  for  a  special  case, 
namely,  for  the  case  in  which  the  moments  of  inertia  of  the  armatures  (or  rotating 
parts)  of  generator  and  synchronous  motor  are  equal  and  for  the  particular  phase 
angle  <£  =  180°,  see  Figs.  149  and  152.  In  this  case,  namely,  when  0  =  180°,  a  small 
change  of  Pf  is  accompanied  by  an  equal  and  opposite  changfe  of  Pffy  so  that  equal 
unbalanced  torques  act  at  each  instant  on  the  armatures  of  machines  A  and  £,  and 
their  moments  of  inertia  being  equal  the  ranges  of  the  oscillations  of  the  armatures  of 
both  machines  are  equal.  That  is,  the  armature  of  machine  A  is  as  much  ahead 
of  its  mean  position  at  each  instant  as  the  armature  of  machine  B  is  behind  its  mean 
position  at  the  same  instant  and  vice  versa.  Therefore  the  change  of  the  phase  angle 
</>  is  equal  to  2/i/>,  2i/>  being  the  angular  displacement  of  one  armature  referred  to  the 
other  and  /  being  the  number  of  pairs  of  field  magnet  poles  in  each  machine. 


200  ELEMENTS    OF   ALTERNATING   CURRENTS. 

Differentiating  equation  (89)  with  respect  to  0,  writing  2/i/>  for  d$^  and  after  the 
differentiation  is  performed,  putting  0  =  180°,  we  have 


Now  dP"  equals  2-xn  T  where  n  is  the  speed  of  the  machine  and  T  is  the  unbal- 
anced torque.     Therefore 


The  value  of  b,  equation  (i),  is  therefore 

pAB 

b  =. -4—          —  •  sm  6 

irnV  R*  -f  w2Z2 

Substituting  this  value  of  b  in  equation  (ii)  and  solving  for  /2,  we  have 


2  _ 


pAB  sin  0 
or,  since 

.     ,  uL 

sin  v  —  -     and     co  = 

V  R* 
we  have 


130.  The  efficiency  of  transmission  of  power  by  means  of  an 
alternating  generator  and  a  synchronous  motor,  —  The  ratio, 
P"/Pf,  of  the  motor  intake  to  the  generator  output  is  called  the 
efficiency  of  transmission.  This  is  not  the  net  efficiency  of  the 
system  inasmuch  as  power  consumed  in  field  excitation  of  the  two 
machines  and  power  lost  by  friction  and  by  eddy  currents  and 
hysteresis  are  not  considered.  The  conditions  necessary  for 
maximum  efficiency  of  transmission  depend  upon  which  of  the 
quantities  A,  B,  a>L,  R  and  P"  are  open  to  choice  or  capable  of 
adjustment.  The  quantities  wL  and  R  are  ordinarily  fixed  in 
value,  while  A  and  B  may  be  changed  more  or  less  by  varying 
the  field  excitation  of  the  respective  machines,  and  P"  may  be 
varied  by  changing  the  load  on  the  motor. 

I.  When  the  electromotive  force  A  of  the  generating  alternator  is 
adjustable,  maximum  efficiency  is  obtained  when  the  current  (and 
also  Rf2)  is  a  minimum  ;  values  of  B,  coL,  R  and  P"  being  given. 
This  minimum  current  is  obtained  when  A  is  adjusted  until  B 
and  /  are  opposite  to  each  other  in  phase. 


THE   SYNCHRONOUS   MOTOR.         .  2OI 

Proof. — The  motor  intake  is  P"  =  BI  cos  (BI),  according  to 
equation  (86).  Therefore,  since  P"  and  B  are  given,  the  minimum 
value  of  7  corresponds  to  maximum  value  of  cos  (BI).  But  the 
maximum  (negative)  value  of  cos  (BI)  is  —  1,  and  the  corre- 
sponding value  of  the  angle  (BI)  is  180°. 

To  calculate  the  value  of  A  which  will  bring  B  and  7  opposite 
to  each  other  in  phase  for  the  given  values  of  B,  coL,  R  and  P" y 


Fig.  157. 


consider  Fig.  157,  in  which  7  represents  the  current,  and  E  the 
resultant  electromotive  force  of  machines  A  and  B.  From  this 
figure  we  have 


from  which  A  may  be  calculated  when  7  is  known.  The  value 
of  7  may  be  determined  from  the  equation  P"  =  BI  cos  (BI). 

2.  When  the  electromotive  force  B  of  the  synchronous  motor  is 
adjustable,  maximum  efficiency  is  obtained  when  the  current  (and 
also  RI2)  is  a  minimum  ;  values  of  A,  o)L,  R  and  P"  being  given. 
This  minimum  current  is  obtained  when  B  is  adjusted  until  A 
and  7  are  in  phase  with  each  other.  The  proof  of  this  proposi- 
tion is  given  below. 

To  calculate  the  value  of  B  which  will  bring  A  and  7  into 
phase  with  each  other  for  the  given  values  of  A,  &L,  R  and  P", 
consider  Fig.  158,  in  which  A  represents  the  value  of  the  elec- 


Fig.  158. 


202 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


tromotive  force  of  machine  A,  and  E  represents  the  resultant 
electromotive  force  of  machines  A  and  B.     From  this  figure  we 

have 

(A  - 


from  which  B  may  be  calculated  when  /  is   known.     The  value 
of  /  may  be  calculated  from  the  relation 

P"  +  RP 


Proof  that  I  is  a  minimum  when  B  is  adjusted  until  A  and  I  are 
in  phase  with  each  other  ;  A,  coL,  R  and  P"  being  given.  —  Plot 
the  curve  of  which  the  ordinates  represent  values  of  coLI  and  the 
abscissas  represent  values  of  B  cos  (BI).  This  curve  is  a  hyper- 


Fig.  159. 

bola  inasmuch  as  B  cos  (BI)  x  /=  P"  or  B  cos  (BI)  x  toLI  = 
<t)LP"  =  a  constant.  This  curve  is  plotted  in  Fig.  159  with  the 
values  of  B  cos  (BI)  laid  off  to  the  left  inasmuch  as  P"t  being 
an  intake,  is  negative.  The  ;tr-axis  of  reference  OIt  represents 
the  current  in  the  circuit ;  and  the  line  OEt  making  with  01  the 
known  angle  0,  but  of  unknown  length,  represents  the  resultant 
electromotive  force  of  machines  A  and  B. 

Draw  a  horizontal  line  cutting  OE  in  p  and  cutting  the  hyper- 
bola in  q.     Then  for  that  particular  value  of  the  current  which 


THE   SYNCHRONOUS   MOTOR. 


203 


corresponds  to  the  chosen  ordinate  aq  (  =  e»Z7)  the  line  Op  rep- 
resents the  actual  resultant  electromotive  force  inasmuch  as  the 
vertical  component  E  is  equal  to  a>LI.  Describe  about  the  point 
/  a  circle  of  which  the  radius  represents  A.  Then  the  lines  Or 
and  Os  represent  the  two  possible  values  of  B  for  the  chosen 
value  of  a>LI.  From  this  diagram  it  is  evident  that  for  the  short- 
est possible  value  of  Op  (smallest  possible  current)  the  circle  de- 
scribed about/ just  touches  the  ordinate  aq,  in  which  case  the 
line  A  is  horizontal  and  parallel  to  01,  as  shown  in  Fig.  160. 


Fig.  160. 

For  any  shorter  value  of  Op  (smaller  value  of  E  or  7)  the  circle 
does  not  reach  to  the  ordinate  aq,  which  means  that  for  so  small 
a  value  of  the  current  the  given  value  of  A  is  too  small  to  supply 
the  line  losses  Rf2  and  the  given  motor  intake  P" . 

3.  When  Pn  is  adjustable,  maximum  efficiency  occurs  when 
the  differential  coefficient  of  PffjP'  with  respect  to  0  is  zero. 
From  equations  (89)  and  (90)  we  have 

P"      AB  cos  (<ft  -  0)  +  &  cos  0 
P7  =  AB  cos  (4>  +  ff)  +  A2  cos~0 
whence,  applying  the  condition 


=  O 


•Mzv 


204 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


we  have 

A2  sin  (<£  —  0)  —  B2  sin  (<£  +  6)  =  2^4^  sin  0  (93) 

which  determines  the  value  of  </>  for  which  the  efficiency  of  trans- 
mission is  a  maximum. 

Geometrical  construction  of  equation  (93). — Draw  lines,  Fig.  1 6 1 , 

representing  A2  and  B2  to  scale,  the  angle  between  them  being 

^ ^  20.     About  the  point  /  as 

a  center  describe  a  circle  of 
which  the  radius  represents 
2AB  sin  6.  From  the  point 
q  draw  a  tangent  to  this  cir- 
cle. The  angle  between  the 
line  OS  and  this  tangent  is 
"0  the  required  value  of  <£. 
Two  tangents  can  be  drawn 
from  the  point  q.  One  of 
these  tangents  determines 
the  value  of  0  (less  than 

1 80°)  for  which  the  efficiency  of  transmission  is  a  maximum  with 

machine  A  acting  as  a  synchronous  motor  and  the  other  tangent 

determines    the    value    of   <£ 

(greater  than  180°)  for  which 

the  efficiency  of  transmission 

is  a  maximum  with  machine 

B  acting  as  a   synchronous 

motor.      The   machine  A  is 

distinguished  as  having    the 

greater   electromotive   force. 

The  angle  <£  is  the  lag  of  B 

behind  A. 

131.  Value  of  B  to  give 
maximum  intake  of  machine 
B  with  given  current ;  A,  a>L  Fig- 162- 

and  Ry  being  given. — Let  /,  Fig.  162,  be  the  given  current  and 


THE   SYNCHRONOUS   MOTOR.  205 


E  (=  TV ' R1  -f  o)2L2)  the  resultant  electromotive  force.  In  order 
that  the  intake  of  B  may  be  a  maximum,  BI  cos  (BI)  or  -#  cos 
(BI)  must  be  a  maximum.  Now  B  cos  (#/)  is  the  projection  of 
B  on  the  current  line  01.  Describe  a  circle  of  radius  A  about 
the  point  Pt  Fig.  162  ;  then  Ox  is  the  greatest  possible  value  of 
B  cos  (BI)  for  the  given  current  and  OC  is  the  required  value 
of  B.  From  the  triangle  OPC  we  have  as  the  required  value  of  B 


B  =  VA2  +  E2-  2AE  cos  0 

Remark.  —  From  Fig.  162  it  is  evident  that  A  is  in  phase  with 
/when  B  is  adjusted  to  give  maximum  Pn  with  given  /.  It  was 
shown  in  Article  1  30  that  A  is  also  in  phase  with  /  when  B  is 
adjusted  to  give  minimum  /  with  given  Pf  (or  with  given  Pft). 
This  correspondence  is  by  no  means  self-evident. 

132,  Maximum  intake  of  machine  B  ;  A,  B,  &L  and  R  being 
given.  —  P"  has  its  maximum  negative  value  when  cos  (</>  —  6)  — 
—  1  and  equation  (89)  becomes 


™  = 


Fig.  163. 


206  ELEMENTS   OF   ALTERNATING   CURRENTS. 

Fig.  163  shows  the  state  of  affairs  when  intake  of  B  is  at  its 
greatest. 

133.  Greatest  values  of  the  electromotive  force  B  for  which 
machine  B  can  act  as  a  motor ;  A,  coL  and  R  being  given. — So 
long  as 

AB 


is  greater  than 


+ 


cos  0 


then  P"  can  have  negative  values  according  to  equation  (89). 
Therefore  the  limiting  case  is  where 

AB  B2 


cos 


e 


or 


~  cos0 
This  limiting  case  is  shown  in  Fig.  164. 


(95) 


134.  To  find  value  of  B  for  which  the  machine  B  may  take  in 
the  greatest  possible  power  from  A  ;  A,  coL  and  R  being  given. — 


THE   SYNCHRONOUS    MOTOR.  2O/ 

Equation  (94)  expresses  the  greatest  intake  of  B  for  given  values 
of  A,  B,  o)L  and  R.  It  is  required  to  find  the  value  of  B  which 
will  make  this  greatest  intake  a  maximum.  This  value  of  B 
must  render  B2  cos  0  —  AB  [the  numerator  of  right-hand  mem- 
ber of  equation  (94)]  a  maximum.  Differentiating  this  expres- 
sion with  respect  to  B  and  placing  the  differential  coefficient 
equal  to  zero  we  have 

2B  cos  6  —  A  =  o 
or 


Remark.  —  A  comparison  of  equations  (95)  and  (96)  shows 
that  the  value  of  B  for  greatest  possible  intake  of  machine  B  is 
half  the  greatest  value  of  B  for  which  machine  B  can  act  as  a 
motor  at  all.  This  is  also  the  case  with  a  direct-current  motor. 
The  greatest  electromotive  force  such  a  motor  can  have  is  the 
electromotive  force  of  the  dynamo  which  drives  it,  and  the  value 
of  its  electromotive  force  to  permit  the  greatest  possible  intake  is 
one-half  the  electromotive  force  of  the  dynamo  which  drives  it. 

135.  Excitation  characteristics.  —  With  given  load  on  a  synchro- 
nous motor  (given  value  of  Pff)  its  electromotive  force  B  may  be 
changed  by  varying  its  field  excitation,  and  for  each  value  of  B 
there  is  a  definite  value  of  the  current  /.  Thus  the  abscissas  of 
the  curves,  Fig.  165,  represent  values  of  /,  and  ordinates  repre- 
sent values  of  B  for  loads  of  zero,  100  kilowatts  and  200  kilo- 
watts respectively.  These  curves  are  called  the  excitation  charac- 
teristics of  the  motor.  Fig.  165  is  based  on  the  values  A  =  1,100 
volts,  R  =  i  ohm  and  coL  =  0.58  ohm.  For  the  greatest  possi- 
ble intake,  302.7  kilowatts,  the  characteristic  reduces  to  the  point 
enclosed  in  the  small  circle.  It  was  pointed  out  in  Article  127 
that  with  given  load  there  are  two  values  of  /  for  each  value  of 
B,  and  that  the  larger  value  of  /  corresponds  to  unstable  and  the 
smaller  value  to  stable  running.  The  dotted  portions  of  the 
curves,  Fig.  165,  correspond  to  the  larger  values  of  /.  These 


208 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


(Zoo 


Hoc 


JOo 


800 
7oo 

600 

5oo 

+00 

£00 
loo 


CD 


fcOOK, 


X^ 


0K. 


2.7  K.W. 


N&: 


AMPERES 


\ 


\ 


O  loo        Zoo      Joo       -foe       600       t*o       Too       800       900       lOoo      UQQ 

Fig.  165. 

dotted  portions  cannot,  of  course,  be  determined  by  experiment, 
on  account  of  the  instability  of  running. 

The  equation  to  the  excitation  characteristics  may  be  derived  as  follows  :  Let  /, 
Fig.  166,  represent  the  current  and  E  the  resultant  electromotive  force ;  the  compo- 


£  O       RI 

Fig.  166. 

nents  of  E  are  R I  and  uLI.     The  electromotive  force  E  is  the  vector  sum  of  A  and  B, 

F* 

as  shown,  and  the  component  of  B  parallel  to  /  is  —  -    From  the  right-angled  tri- 
angles of  the  figures  we  have 


THE   SYNCHRONOUS    MOTOR. 


209 


By  eliminating  jc  from  these  equations  we  have  the  required  relation  between  B 
and  /;  /)//,  A,  R  and  uL  being  given.  The  curves,  Fig.  165,  were  calculated 
graphically  by  means  of  the  diagram,  Fig.  149. 

136.  The  negative  reactance  of  the  over-excited  synchronous 
motor. — When  the  electromotive  force  of  a  synchronous  motor 
is  greater  than  the  electromotive  force  of  the  generator  which 
drives  it,  the  motor  is  called  an  over-excited  synchronous  motor. 
Such  a  synchronous  motor  takes  current  which  is  ahead  of  the 
generator  electromotive  force  in  phase,  so  that  an  over-excited 
synchronous  motor  constitutes  a  receiving  circuit  of  negative  re- 
actance like  a  condenser.  This  is  especially  the  case  when  the 
motor  load  is  light.  This  state  of  affairs  is  shown  in  Fig.  167, 
in  which  the  A  machine,  which  has  the  greater  electromotive 
force,  is  the  motor  and  is  at  zero  load. 

To  calculate  the  effectiveness  of  an  over-excited,  light-running 
synchronous  motor  as  a  compensator  for  lagging  currents  when 
connected  in  parallel  with  an  inductive  receiving  circuit,  repre- 
sent by  the  line  A,  Fig. 
167,  the  electromotive 
force  of  the  synchronous 
motor,  by  B  the  electro-  , 
motive  force  between  the 
mains  at  the  receiving 
station,  the  angle  0  being  ^ 
determined  by  the  resis- 
tance and  reactance  of 
the  armature  of  the  syn- 
chronous motor.  The 
component  of  /  which  is 

n       i  i       r    r>   •  1_  Fig-  167. 

90    ahead  of  B  in  phase 

is  /  sin  (f>,  where  cf>  is  the  angle  at  C  of  the  triangle  OCP  and  / 
is  equal  to  the  resultant  electromotive  force  OP  divided  by  the 
impedance  of  the  motor. 


210  ELEMENTS   OF   ALTERNATING   CURRENTS. 

137.  The  polyphase  synchronous  motor. — The  preceding  arti- 
cles refer  explicitly  to  the  single-phase  synchronous  motor,  that 
is,  to  a  single -phase  alternator  taking  power  as  a  motor  from  a 
single-phase  generator.     The  entire  discussion   applies  equally 
well,  however,  to  the  polyphase  synchronous  motor,  that  is,  to  a 
polyphase  alternator  taking  power  as  a  motor  from  a  polyphase 
generator.     In  this  case   each  armature  winding  of  the  motor 
takes  current  from  one  phase  (or  one  armature  winding)  of  the 
generator,  and   the   total    power  intake  is  nPn ',  where  n  is  the 
number  of  phases  and  P"  is  the  intake  of  each  phase.     When 
the  preceding  discussion  is  applied  to  the  polyphase  synchronous 
motor,  the  letters  A,  B,  a>L,  P' ,  P"  and  /  refer  to  one  phase  only 
of  the  system. 

138.  The  rotary  converter. — The  next  chapter  is  devoted  to 
the  rotary  converter,  which,  as  ordinarily  used,  is  at  once  a  syn- 
chronous alternating-current  motor  and  a  direct-current  dynamo. 
This  machine  usually  runs  as  a  polyphase  synchronous  motor, 
taking  power  from  a  polyphase  alternating-current  generator  and 
giving  out  power  in  the  form  of  direct  current.     When  run  in 
this  way  the  rotary  converter  exhibits  all  the  properties  of  the 
synchronous  motor  as  outlined  in  the  foregoing  articles.     The 
next  chapter  is  devoted  mainly  to  the  discussion  of  the  relations 
between  the  direct-current  output  and  the  alternating-current  in- 
take, to  the  relations  between  the  direct  electromotive  force  and 
the  alternating  electromotive  force  of  the  rotary  converter  and 
to  armature  heating. 

PROBLEMS. 

95.  Plot  curves  showing  the  values  of  P',  P"  and  RP  for  two 
synchronous  alternators  in  series,  the  one  alternator  A  having  an 
electromotive  force  of  1,200  volts,  the  other  machine  B  an  elec- 
tromotive force  of  800  volts,  the  resistance  of  the  circuit  being 
I  ohm  and  the  reactance  2  ohms.  Result  similar  to  Fig.  151. 


THE   SYNCHRONOUS    MOTOR.  211 

96.  Two  governed  engines  drive   two  alternators  which  are 
connected  in  multiple  and  feed  one  pair  of  mains.     Engine  A 
drops  from  150  revolutions  per  minute  at  zero  load  to  145  revo- 
lutions per  minute  at   100  horse  power.     Engine  B  drops  from 
150  revolutions  per  minute  at  zero  load  to   147  revolutions  per 
minute  at  full  load  of  75  horse  power.     The  station  output  is 
such  as  to  require  a  total  of  1 2  5  horse  power  to  be  delivered  by 
the   engines.     What  power  is   delivered   by   each   engine,   and 
what  is  their  common  speed  ?     What  power  is  delivered  by  en- 
gine B  when  engine  A  is  idle?     Ans.  (a)  70.2  H.  P.,  (U)  54.8  H. 
P.,  (e)  18.7  H.  P. 

97.  An  alternator  B,  of  which  the  electromotive  force  is  1,000 
volts,  takes   75    kilowatts   from  alternator  A.     To  what  value 
must  the  electromotive  force  of  A  be  adjusted,  so  that  the  effi- 
ciency of  transmission  may  be  maximum,  and  what  is  the  corre- 
sponding value  of  the  current,  resistance  of  circuit  being  I  ohm 
and  reactance  of  circuit  being  0.58  ohm?     Ans.  1,076  volts,  75 
amperes. 

98.  The  electromotive  force  of  alternator  A  in  problem  97  is 
1,076  volts.     To  what  value  must  the  electromotive  force  of  al- 
ternator B  be  adjusted  to  give  maximum  efficiency  of  transmis- 
sion, and  what  is  the  corresponding  value  of  the  current,  intake 
of  B  being  75  kilowatts?     Ans.  1,009.8  volts,  74.9  amperes. 

99.  The  electromotive  force  of  alternator  A  is  1,100  volts,  and 
that  of   alternator  B    is    1,000  volts,  resistance  of   circuit  is   I 
ohm  and  reactance  of  circuit  0.58  ohm.     What  is  the  angular 
lag  of  B  behind  A,  for  which  power  is  most  efficiently  trans- 
mitted from  A  to  B,  what  is  the  intake  of  B  under  these  condi- 
tions, what  is  the  output  of  A,  and  what  is  the  efficiency  ?     Ans. 
181.8°,  P"  =  —  92  kilowatts,  Pr  =  -f  101.8  kilowatts,  efficiency 
91.3  per  cent. 

100.  What  is  the  angular  lag  <f>  of  B  behind  A,  in  problem  95, 
for  which  power  is  most  efficiently  transmitted  from  B  to  A  ;  what 
is  the  intake  of  A  under  these  conditions,  what  is  the  output  of 


212  ELEMENTS   OF   ALTERNATING   CURRENTS. 

B,  and  what  is  the  efficiency?     Ans.  159.4°,  Pr  =  —  36.4  kilo- 
watts, P"  =  -f  156.7  kilowatts,  efficiency  23.2  per  cent. 

101.  The  electromotive  force  of  A  is  900  volts,  that  of  B  is 
800  volts,  resistance  of  circuit  is  I  ohm,  and  reactance  of  circuit 
is  i  ohm.     What  is  the  maximum  intake  of  machine  B  as  a  syn- 
chronous motor  ?     Ans.    1 89. 1  kilowatts. 

102.  What  is  the  greatest  value  of  B  (A,  (oL,  and  R  being  as 
in  problem  97)  which  will  permit  machine  B  to  act  as  a  synchro- 
nous motor  ?     Ans.  1,272  volts. 

103.  What  value  must  B  have  in  order  that  the  maximum  in- 
take of  machine  B  may  be  the  greatest  possible,  and  what  is  this 
intake?     Ans.  636  volts,  220.3  kilowatts. 

104.  Given  A  =  800  volts,  R  =  I  ohm,  coL  =  I  ohm,  P"  = 
30  kilowatts  ;  plot  curve  showing  different  values  of  /correspond- 
ing to  different  values  of  B.     Result  similar  to  Fig.  165. 

105.  An   alternator  of  which  the  electromotive  force   is   150 
volts  is  to  be  run  as  a  motor  from   I  lo-volt  mains.     The  resist- 
ance of  the  alternator  armature  is  I  ohm.     What  is  the  minimum 
amount  of  inductance  required  in  the  circuit,  the  frequency  being 
60  cycles  per  second  ?     Ans.  0.00246  henry. 

1 06.  A  condenser  is  connected  in  series  with  alternators  A  and 
B  so  that  the  total  reactance  of  the  circuit  is  —  0.58  ohm.     The 
resistance  of  the  circuit  is    I  ohm,   the  electromotive  force  of 
machine  A  is  I ,  I  oo  volts,  and  the  electromotive  force  of  machine 
B  is  1,000  volts.     Plot  curves  showing  the  values  of  Pf,  Pn  and 
RI2  for  various  values  of  <£. 

107.  An  alternator  A  has  an  electromotive  force  of  1 , 100  volts, 
a  resistance  of  i  ohm,  and  a  reactance  of  0.58  ohm.     The  machine 
is  driven  as  a  synchronous  motor  with  zero  load  from  i,ooo-volt 
mains.     What  is  the  value  of  the  current  ?     What  is  the  com- 
ponent of  this  current  which  is  90°  ahead  of  the  supply  electro- 
motive force  in  phase  ?     What  capacity  of  condensers  would  take 
the  same  amount  of  leading  current  from  the  i,ooo-volt  mains  at 
a  frequency   of  60   cycles   per  second?     Ans.    268,7  amperes, 
253.1  amperes,  4,574  microfarads. 


CHAPTER  XIII. 

THE  ROTARY  CONVERTER. 

139.  The  rotary  converter. — An  ordinary  direct-current  dynamo 
may  be  made  into  an  alternator  by  providing  it  with  collecting 
rings,  as  described  below,  in  addition  to  its  commutator.  Such 
a  machine  is  called  a  rotary  converter. 

The  single-phase  converter  is  provided  with  two  collecting  rings 
which,  in  case  of  a  two-pole  machine,  are  connected  to  diametric- 
ally opposite  armature  conductors. 

The  two-phase  converter  is  provided  with  four  collecting  rings 
which,  in  case  of  a  two-pole  machine,  are  connected  to  armature 
conductors  90°  apart. 

The  three-phase  converter  is  provided  with  three  collecting  rings 
which,  in  case  of  a  two-pole  machine,  are  connected  to  armature 
conductors  120°  apart. 

Remark  i. — It  is  often  convenient  to  refer  to  a  rotary  converter 
as  a  two-ring,  three-ring,  four-ring,  or  ;/-ring  converter,  as  the 
case  may  be. 

Remark  2. — In  case  of  a  multipolar  machine  the  n  collecting 
rings  are  connected  to  the  armature  as  follows  :  Ring  No.  I  is 
connected  to  all  armature  conductors  which,  for  any  given  posi- 
tion of  the  armature,  lie  midway  under  the  north  poles  of  the 
field  magnet.  Let  /  be  the  distance  between  adjacent  conductors 
of  this  first  set,  that  is,  the  distance  from  north  pole  to  the  next 
north  pole.  Then  ring  No.  2  is  connected  to  the  armature  con- 
ductors which  are  — th  of  /  ahead  of  the  first  set ;  ring  No.  3  is 

connected  to  the  armature  conductors  which  are  — ths  of  /  ahead 

n 

213 


214 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


of  the  first  set ;  ring  No.  4  is  connected  to  the  conductors  which 

are  — ths  of  /  ahead  of  the  first  set,  and  so  on.     This  statement 
n 

applies  to  multicircuit  winding.  In  case  of  the  two-circuit  wind- 
ing each  collecting  ring  is  connected  to  one  armature  conductor 
only.  Fig.  168  shows  a  four-pole  dynamo  with  two  collecting 
rings  each  connected  to  two  armature  conductors.  The  machine 
when  provided  with  these  collecting  rings  is  a  four-pole  single- 
phase  rotary  converter. 

Use  of  the  rotary  converter. — The  rotary  converter  may  be  used 
as  an  ordinary  direct-current  dynamo  or  motor ;  as  an  alternator 
or  synchronous  motor ;  it  may  be  driven  as  a  direct-current 
motor,  the  load  being  provided  by  taking  alternating  current  from 
its  collecting  rings  ;  or  it  may  be  driven  as  a  synchronous  alter- 
nating-current motor,  the  load  being  provided  by  taking  direct 
current  off  the  commutator.  This  last  is  the  principal  use  of  the 

machine.  In  most  cases 
where  power,  transmit- 
ted to  a  distance  by 
alternating  current,  is  to 
be  used  in  the  form  of 
direct  current,  the  rotary 
converter  is  used  for 
bringing  about  the  con- 
version from  alternating 
current  to  direct  current. 
Thus,  in  many  extended 
electric  railway  plants, 
it  is  found  expedient  to 
transmit  the  power  as 
high  pressure  poly- 
phase current  from  a  central  station  to  rotary  converters  stationed 
along  the  line  of  the  railway;  these  rotary  converters  take  the  alter- 
nating current  through  step-down  transformers  and,  in  their  turn, 
supply  direct  current  at  medium  pressure  to  the  trolley  wires. 


THE   ROTARY    CONVERTER. 


215 


140.  The  starting  of  the  rotary  converter  and  its  operation  when 
nsed  to  convert  alternating  current  into  direct  current. — When 
used  in  this  way  the  rotary  converter  is  a  synchronous  motor  and 
it  differs  but  little  in  its  operation  from  the  synchronous  motor 
with  a  belt  load. 

Starting. — The  machine  may  be  started  as  a  direct-current 
motor  using  storage  batteries  or  other  local  source  of  direct  cur- 
rent ;  or  it  may  be  started  in  precisely  the  same  manner  as  a 
synchronous  motor  with  a  belt  load  as  described  in  Article  1 26. 
The  field  magnet  of  a  rotary  converter  is  always  excited  by  direct 
current  taken  from  the  machine  itself. 

Operation. — Let  B,  Fig.  169,  be  the  effective  alternating  elec- 
tromotive force  of  a  rotary  converter  and  A  the  electromotive 
force  of  the  alternating  generator.  Fig.  169  is  identical  to  Fig. 


153,  Chapter  XII.  When  no  direct  current  is  taken  from  the 
converter  its  load  is  zero  and  the  point  Pt  Fig.  169,  is  at  s. 
When  direct  current  is  taken  from  the  converter  the  point  P 
moves  towards  M.  The  alternating  current  taken  by  the  con- 
verter, being  proportional  to  the  resultant  electromotive  force 


2l6  ELEMENTS   OF   ALTERNATING   CURRENTS. 

OP,  at  first  decreases  and  then  increases  *  as  the  direct  current 
load  increases,  and  so  on,  exactly  as  in  the  case  of  a  synchro- 
nous motor  with  a  belt  load.  (See  Article  122.) 

It  was  pointed  out  in  Chapter  XII.  that  a  synchronous  motor 
may  operate  at  comparatively  high  efficiency  for  a  wide  range  of 
values  of  B  (value  of  A  given)  if  the  alternating-current  circuit 
has  large  reactance ;  in  fact,  B  may  even  be  larger  than  A,  as 
pointed  out  in  Article  133.  If  a  considerable  portion  of  the  re- 
actance is  external  to  the  armature  of  the  converter  then  the 
electromotive  force  between  the 'collecting  rings  of  the  converter 
changes  with  B  and  so  also  does  the  electromotive  force  of  the 
machine.  Therefore  the  direct  electromotive  force  of  a  rotary 
converter  may  be  varied  at  will  f  by  changing  the  field  excitation 
of  the  machine,  although  the  electromotive  force  A  of  the  alter- 
nating generator  may  be  constant. 

Hunting. — The  hunting  action  of  the  sychronous  motor  is  de- 
scribed in  Article  1 29.  The  synchronous  motor  with  a  belt  load 
does  not  often  hunt,  for,  the  inelasticity  of  the  belt,  the  slack  in 
the  belt,  and  the  friction  of  the  driven  machinery  tend  to  steady 
the  machine.  The  rotary  converter,  on  the  other  hand,  is 
peculiarly  subject  to  hunting,  and  if  the  pulsations  of  the  engine, 
which  drives  the  alternator,  are  in  unison  with  the  hunting  oscil- 
lations of  the  converter  and  alternator,  hunting  is  sure  to  be  pro- 
duced. A  method  for  damping  the  hunting  oscillations  of  a 
rotary  converter  is  described  in  Article  142. 

141.  Preliminary  statement  concerning  armature  current  of  a 
rotary  converter. — Consider  a  given  armature  conductor  of  a 
rotaiy  converter.  A  part  of  the  current  in  this  conductor  is  due 
to  the  alternating  currents  which  flow  into  the  armature  at  the 
collecting  rings  and  a  part  is  due  to  the  direct  current  flowing 
out  of  the  armature  at  the  direct  current  brushes.  The  actual 
current  in  the  conductor  is  the  algebraic  sum  of  these  two  parts, 

*  When  B  is  less  than  A. 

fThe  possible  range  of  variation  depends  upon  the  reactance  in  the  circuit  exter- 
nal to  the  rotary  converter. 


THE    ROTARY   CONVERTER. 


and  since  these  parts  are  generally  opposite  in  sign,  therefore  the 
actual  current  in  the  conductor  is  rather  small  and  so  also  is  its 
magnetic  effect  and  its  heating  effect. 

142.  Magnetic  reaction  of  the  armature  of  the  rotary  converter. 

Distortion  of  field. — The  distortion  of  the  magnetic  field  of  a 
dynamo  by  the  armature  currents  accompanies,  and  is  in  fact  the 
cause  of,  the  torque 
with  which  the  field 
acts  upon  the  armature. 
When  the  torque  is  in 
the  direction  of  the  ro- 
tation of  the  armature 
(motor  action)  the  field 
is  concentrated  under 

the  leading  horns  of  the  FI§.  170. 

pole  pieces  as  shown  in  Fig.  1 70.  When  the  torque  is  opposite  to 
speed  the  field  is  concentrated  under  the  trailing  horns  of  the  pole 

pieces  as  shown  in  Fig.  171. 
When  a  rotary  converter 
is  running  steadily  the  speed 
of  its  armature  is  constant, 
and  the  only  torque  acting  on 
the  armature  is  the  slight 
torque  needed  to  overcome 

__    friction,  therefore  the  field  is 

Fie- 17L  scarcely  at  all  distorted. 

When  a  rotary  converter  hunts,  its  speed  oscillates  above  and 
below  synchronism  so  that  a  torque  acts  upon  the  armature,  first 
in  one  direction  and  then  in  another,  and  the  field  is  concen- 
trated, first  under  the  trailing  horns  and  then  under  the  leading 
horns  of  the  pole  pieces. 

The  damping  of  the  hunting  oscillations  of  a  rotary  converter  is 
accomplished  by  a  heavy  copper  frame  ccc,  Fig.  172,  which  is 
embedded  in  the  face  of  each  field  pole  as  shown.  The  shifting 


218 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


B 


of  the  flux  from  one  side  to  the  other  of  the  pole  face,  as  de- 
scribed above,  induces  large  currents  in  the  circuits  A  and  B, 
Fig.  172.  These  currents  oppose  the  shifting  of  the  flux  and 
thereby  damp  the  oscillations. 

Demagnetizing  action. — The  demagnetizing  action  of  the  arma- 
ture currents  of  a  rotary  converter  may  be  considered  as  made  up 
of  the  demagnetizing  action  of  the  direct  current  alone  and  of  the 

alternating  currents  alone.  The  first 
is  the  same  as  in  the  direct- current 
dynamo  and  the  second  is  consid- 
ered in  Article  79. 

A  very  important  effect  due  to 
the  demagnetizing  action  of  the  ar- 
mature currents  in  a  rotary  con- 
verter is  the  following.  A  converter 
takes  direct  current  from  constant 
electromotive  force  mains  and  deliv- 
ers alternating  current.  When  a 
short  circuit  occurs  on  the  alternat- 
ing current  mains  the  converter 
speeds  up  indefinitely  and  may  be 
destroyed.  The  enormous  lagging 
alternating  current  weakens  the  field  and,  since  but  little  mechan- 
ical effort  is  required,  the  speed  increases  just  as  the  speed  of  an  un- 
loaded direct-current  motor  would  increase  by  weakening  its  field. 

143.  Power  rating  of  rotary  converters. — The  magnetic  action 
(demagnetizing  action  and  distorting  action)  of  the  armature  cur- 
rents of  a  rotary  converter  is  never  troublesome,  so  that  the  al- 
lowable output  is  limited  by  the  permissible  heating  of  the  arma- 
ture. The  armature  heating  is  rather  small,  as  pointed  out  in 
Article  141,  so  that  a  given  machine  has  a  higher  power  rating  as 
a  rotary  converter  than  as  a  direct-current  dynamo,  except  in  the 
case  of  the  single-phase  converter.  The  accompanying  table 
gives  the  power  ratings  (based  upon  equal  average  armature 


Fig.  172. 


THE    ROTARY   CONVERTER.  219 

heating)  of  a  given  machine  when  used  (a)  as  a  direct-current 
dynamo,  (b)  as  a  single-phase  converter,  (c)  as  a  three-phase  con- 
verter, (d)  as  a  two-phase  (four-ring)  converter,  and  (e)  as  a  six- 
phase  converter. 

POWER  RATINGS  OF  ROTARY  CONVERTERS.* 
a.                        b.                        c.                      d.  e. 
Continuous-            Single-             Three-               Four-  Six- 
current                 phase                phase                 ring  phase 
dynamo.            converter.         converter.         converter.  converter. 

I. oo  .85  1.32  1.62  1.92 

144.  Electromotive  force  relations  of  the  rotary  converter. — Let 

EQ  be  the  electromotive  force  between  the  direct-current  brushes 
and  En  the  effective  alternating  electromotive  force  between  ad- 
jacent collecting  rings  of  an  w-ring  converter.  The  ratio  -~-  has 

a  characteristic  value  for  each  value  of  #. 

Fundamental  assumption. — Consider  an  armature  conductor  c, 
Fig.  173,  at  angular  distance  ft  from  the  axis  of  the  field,  as 


axis  oi 
"Hell"" 


Fig.  173. 

shown,  f     We  assume  that  the  electromotive  force  induced  in  the 
conductor  c  is  proportional  to  cos  ft  or  equal  to  C  cos  ft  where 

*  These  ratings  are  calculated  as  explained  in  Article  147,  and  in  their  calculation 
the  losses  in  the  machine  and  the  wattless  component  of  the  alternating  currents  have 
been  ignored.  These  ratings  are  therefore  somewhat  too  large. 

f  The  discussion  in  Articles  144  to  147  is  given  for  the  case  of  a  two-pole  machine. 
The  results,  however,  apply  to  multipolar  machines  as  well. 


220  ELEMENTS   OF   ALTERNATING   CURRENTS. 

C  is  a  constant.  The  results  of  this  assumption  are  practically 
in  accord  with  experiment. 

The  number  of  armature  conductors  between  c  and  c'  is  pro- 
portional to,  or  say  equal  to,  d(3. 

The  electromotive  force  in  each  conductor  is  C-cos  /3,  and  the 
electromotive  force  in  all  the  conductors  between  c  and  c'  is  : 

de=C-cos$-dl3.  (a) 

Electromotive  force  EQ  between  direct  current  brushes. — All  the 
conductors  between  b  and  b'  are  in  series  between  the  direct-cur- 
rent brushes  so  that  • 


or 

E,=  2C  (b) 

Effective  electromotive  force  En  between  adjacent  collecting  rings 
of  an  n-ring  converter. — The  electromotive  force  between  adjacent 
rings  r  and  r' ,  Fig.  174,  is  at  its  maximum  value  when  the  arc 


Fig.  174. 


rr1  is  bisected  by  the  axis  of  the  field  as  shown.  The  angle  be- 
tween r  and  r'  is  —  or  half  this  angle  is  -.  The  maximum 
electromotive  force  ^  2En  between  rings  r  and  r'  is  therefore  : 


/• 
= 
J— 


or  since  2  C  =  £Q,  we  have 

£H  =  —  EQ  sin  7T/n  (97) 


THE   ROTARY    CONVERTER.  221 

Examples.  —  The  effective  alternating  electromotive  force  of  a 
single-phase  converter  (n  =  2)  is  : 

£-^-.7073,      '•  '.'      (98) 

The  effective  alternating  electromotive  force  between  adjacent 
rings  of  a  three-phase  converter  (n  =  3)  is  : 

(99) 

The  effective  alternating  electromotive  force  between  adjacent 
rings  of  a  two  -phase  converter  (n  =  4)  is  : 


The  effective  electromotive  force  between  opposite  rings  of  a 
two  -phase  converter  is  E2. 

145.  Current  relations  of  the  rotary  converter.  Fundamental 
assumptions.  —  In  the  discussions  of  current  relations  we  shall  as- 
sume that  the  alternating  current  flowing  through  each  section 
(between  adjacent  collecting  rings)  of  the  armature  is  exactly 
opposite  in  phase  to  the  alternating  electromotive  force  in  that 
section,  and  that  the  intake  and  output 
of  power  are  equal. 


Let  70  be  the  output  of  direct  current 


and  let  7n  be  the  effective  alternating 

current  flowing  in  the  armature  between 

two  adjacent  collecting  rings.     The  in- 

take of  power  per  phase  is  EJn  or  the  total  intake  is  nEJn  and  the 

power  output  is  EQTQ.     Therefore 

y.-«v. 

or  substituting  the  value  of  En  from  equation  (97)  we  have  : 


/.—        V  (10.) 

n  sm  7rn 


222  ELEMENTS    OF   ALTERNATING   CURRENTS 

Ctirrent  in  each  main.  —  The  current  /  in  each  main  or  the 
current  entering  the  armature  at  each  collecting  ring,  is  the  vector 
difference  between  //  and  //',  Fig.  175,  so  that 

/  =  2/n  sin  7T/n  (IO2) 

Examples.  —  The  effective  alternating  current  in  each  half  of 
the  armature  of  a  2  -pole  single-phase  converter  (n  —  2)  is  : 

/2=^  (.03) 

and  the  effective  alternating  current  entering  at  each  collecting 
ring  is 

/  =  2/2=  1.414/0  (I04) 

The  effective  alternating  current  flowing  in  the  armature  be- 
tween adjacent  collecting  rings  of  a  2  -pole  three-phase  converter 
(n  =  3)  is  : 

.  .005) 


and  the  effective  current  entering  at  each  collecting  ring  is 

/=^3/3  (106) 

The  effective  alternating  current  flowing  in  the  armature  be- 
tween adjacent  collecting  rings  of  a  2  -pole  two  -phase  converter 
(n  =  4)  is  : 


(107) 

and  the  effective  current  entering  at  each  collecting  ring  is 

/r=*/2/4  (1  08) 

146,  Instantaneous  current  in  a  given  armature  conductor  of  a 
rotary  converter.  —  Letr  and  rf  ',  Fig.  176,  be  the  points  of  attach- 
ment of  adjacent  collecting  rings  of  an  ^-ring  converter  and  let 
the  line  0  M  bisect  the  arc  rr'  .  Consider  an  armature  conductor 
c  between  r  and  r'  and  let  the  angle  cOM  be  represented  by  a. 
The  largest  possible  value  of  a  is  TT/H  or  one-half  of  the  angle 
between  r  and  r'  . 


THE    ROTARY   CONVERTER. 


223 


Let  cot  be  the  angle  between   OM  and  the  axis  of  the  field. 
Then  the  alternating  current  between  the  collecting  rings  r  and 


Fig.  176. 

r*t  that  is,  the  alternating  current  in  the  conductor  c,  is  at  its 
maximum  value  >/2/n  when  tot  =  o.  Therefore  the  expression 
for  the  instantaneous  value  of  this  current  is  ^2/B  cos  at. 

When  the  conductor  c  is  at  brush  b' ,  cot  =  90°  —  a,  when  the 
conductors  reaches  brush  b,  cot  =  270°  —  a,  and  when  conductor 
c  reaches  brush  b'  again  cot  =  450°  —  a  and  so  on.  Each  time 

the  conductor  c  passes  a  brush  the  direct  current  — °  (discussion 

applied  to  a  2-pole  machine)  in  conductor  c  is  reversed.  There- 
fore the  total  current  in  conductor  c  is 


i=  v/2/  cos 


(109) 


The  -f  sign  is  to  be  taken  between  cot  =  90°  —  a  and  cot  = 
270°  —  a,  the  —  sign  is  to  be  taken  between  cot  =  270°  —  a  and 
cot  =  450°  —  a,  the  +  sign  again  between  450°  —  a  and  630°  —  a, 
etc. 

Remark. — The  angle  a  determines  simply  the  phase  of  the 
alternating  current  in  the  conductor  at  the  instant  that  the  direct 
current  is  reversed. 


224 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


The  meaning  of  equation  (109)  is  made  more  clear  perhaps 
by  the  curves  shown  in  Figs.  177  (a)  and  (b).  The  upper  curves 
in  each  figure  are  the  component  current  curves  ;  that  is,  the 
ordinates  of  the  square-wave  curve  represent  that  part  of  the  cur- 
rent in  the  given  conductor  which  depends  upon  the  direct  cur- 


Component    curves      7i=2 


Resultant  curve     7i=2     ot=0 

Fig.  177  (a). 


Component  curves    n=.%     oc=  90' 


Resultant  curve    n  = 


Fig.  177(b). 


THE   ROTARY   CONVERTER.  225 

rent  output  of  the  machine,  and  the  ordinates  of  the  sine  curve 
represent  that  part  of  the  current  in  the  given  conductor  which 
depends  upon  the  alternating  current  intake  of  the  machine. 
The  alternating  currents  are  assumed  to  be  exactly  opposite  in 
phase  to  the  alternating  electromotive  forces  of  the  machine. 

147.  The  heating  of  the  armature  conductors  of  a  rotary 
converter. — Equation  (109)  expresses  the  instantaneous  value 
of  the  actual  current  in  a  given  armature  conductor  of  a 
rotary  converter  during  the  time  that  this  conductor  is  pass- 
ing from  one  direct-current  brush  to  the  other ;  that  is,  from 
cof=—  (90°  +  a)  to  ft)/  =  90°  —  a.  The  current  passes  through 
a  similar  set  of  values  during  the  next  half  revolution  of  the 
armature.  The  average  rate  at  which  heat  is  generated  in  the 
given  conductor  is  proportional  to  the  average  value  of  i2,  equa- 
tion (109),  during  the  time  a>/=  —  (90°  -f  a)  to  ft>/=  90°  —  a. 
Therefore 

the  average  rate  of  gener-        / 2  ,  1 6  COS  a  8  \ 

ating  heat  in  the  conduc-  I   I — : ; 1 = — ;— = j-  (  I  I  o) 

tor  c  is  proportional  to  :  4  \  ™  Sin  7T,  n        H2  Sin2  7T/nJ 

The  rate  at  which  heat  would  be  generated  in  a  given  conduc- 
tor by  the  direct  current  alone  would  be  proportional  to  /02/4, 
and  equation  (no)  shows  that  the  conductor  c  has 


/          16  cos  a  8         \ 

\        Trfi  sin  IT jri      n2  sin2  w/nj 


times  as  much  heat  generated  in  it  as  an  //-ring  converter  as 
would  be  generated  in  it  by  the  direct  current  alone. 

The  conductors  midway  between  the  points  of  attachment  of 
the  collector  rings  (a  =  o)  are  heated  least,  and  the  conductors 
near  the  points  of  attachment  of  the  collector  rings  (a  =  ±  ir/n) 
are  heated  most.  For  example,  in  a  two-ring  converter  (n  =  2) 
the  conductors  midway  between  the  points  of  attachment  of  the 
collector  rings  (a  =  o)  have  only  0.453  as  much  heat  generated 
in  them  as  would  be  generated  in  them  by  the  direct  current 
alone,  and  the  conductors  near  the  points  of  attachment  of  the 


226 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


collector  rings  (a  =  ±  90°)  have  three  times  as  much  heat  gener- 
ated in  them  as  would  be  generated  in  them  by  the  direct  cur- 
rent alone. 

The  average  heating  of  the  entire  armature  of  a  rotary  con- 
verter.— The  average  heating  over  the  entire  armature  is 
found  by  integrating  the  equation  ( 1 1  o)  with  respect  to  a  from 

a  =  —  —  to  a  =  -j-  —  and  dividing  the  result  by  — —    This  gives  : 


n 


n 


Average  heating  of  armature  of 
n-ring  converter  is  proportional  to  : 


The  average  heating  is  therefore  I  I  —  —»  -f  -»  — 

\         TT       n2  sm2  7r/ 


)    ti 
nJ 


times 


as  great  as  the  heating  of  the  armature  by  the  direct  current 
alone.     Therefore  an  n-rmg  converter  can  put  out 

i 


I  — 


1.6 


8 


Fig.  178. 


times  as  much  direct  cur- 
rent as  the  same  machine 
can  when  used  as  a  simple 
dynamo,  for  the  same  total 
armature  heating.  The 
table  given  in  Article  143 
is  calculated  in  this  way. 

148.  Use  of  rotary  con- 
verter for  supplying  current 
to  the  Edison  three -wire 
system,— T',  T",  and  T"> ', 
Fig.  178,  represent  the 
three  secondaries  of  the 


step-down  transformer  which  are  Y-connected  to  three  collecting 
rings,  r' ,  /',  and  rf",  of  a  three-ring  converter.      The  electronic- 

*  Generally  less  than  unity. 


THE   ROTARY    CONVERTER. 

tive  force  between  tlie  neutral  point  N  and  either  direct-current 
brush  is  constant  and  equal  to  Jialf  the  electromotive  force  from 
brush  to  brush.  Therefore  the  middle  wire  of  Edison's  three- 
wire  system  may  be  connected  to  N,  the  outside  wires  being  con- 
nected to  the  direct^current  brushes  b  and  b'  . 

Discussion,  —  The  electromotive  force  between  brush  b  and  the 
neutral  point  N  is  the  sum  of  the  electromotive  forces  b  to  r'  and 
r1  to  N.  Let  £Q  be  the  steady  electromotive  force  b  to  b'  ,  and 
let  time  be  reckoned  from  the  instant  that  r'  is  at  b.  Then  the 
electromotive  force  br'  fluctuates  between  zero  (when  t  =  o)  and 
£0,  and  its  instantaneous  value  is 


2  2 

The  effective  electromotive  force  £3  between  rings  is 


(99)bis 

and  the  effective  electromotive  force  in  one  of  the  Y-connected 
units  T',  T",  and  T'"  is  ^3-r-^3  or  EQ  -r-  2\/2,  so  that  the 
maximum  electromotive  force  in  Tf  is  £Q  -T-  2.  Now  the  elec- 
tromotive force  in  T'  is  at  its  maximum  value  when  the  ring  r1  is 
at  b,  that  is,  when  /  =  o  or  when  cot  =  o.  Therefore  the  in- 
stantaneous electromotive  force  in  r'Nis 

~~ 


Therefore 

IP  +  VN=  —  ° 


PROBLEMS. 


1  08.  The  plain  multicircuit  ring-wound  armature  of  a  six-  pole 
direct-current  dynamo  has  360  conductors  on  its  face.  These 
conductors  are  numbered  from  I  to  360. 


228  ELEMENTS   OF   ALTERNATING   CURRENTS. 

(a)  One  collecting  ring  of  a  rotary  converter  is  connected  to 
conductor  No.  i.  To  what  other  conductors  must  this  ring  be 
connected?  Ans.  A  to  121  and  241. 

(fr)  To  what  conductors  must  a  second  ring  (B)  be  connected 
to  give  a  two-ring  converter?  Ans.  B  to  61,  181  and  301. 

(c)  To  what  conductors  must  two  additional  rings  B  and  C  be 
connected  to  give  a  three-ring  converter?  Ans.  B  to  41,  161 
and  281,  C  to  81,  201  and  321. 

(d}  To  what  conductors  must  three  additional  rings  B,  C  and 
D  be  connected  to  give  a  four-ringed  ?  Ans.  B  to  3 1 ,  151  and 
271,  C  to  61,  181  and  301,  D  to  91,  21 1  and  331. 

(e)  To  what  conductors  must  four  additional  rings  B,  C,  D  and 
E  be  connected  to  give  a  five-ring  converter?  Ans.  B  to  25, 
145  and  265,  C  to  49,  169  and  289,  Z>  to  73,  193  and  313,  E 
to  97,  217  and  337. 

109.  A  four-pole,  two-circuit,  single   drum  winding  has   102 
conductors  numbered  consecutively  from   I   to  102.     The  con- 
ductors are  connected  as  follows  :  1-26-51-76-101  .  .  .  28-53—78 
and  back  to  i.     To  what  conductors  must  the  three  rings  A,  B 
and  C  of  a  three-ring  converter  be  connected  ?     Ans.  A  to  No. 
i,  B  to  No.  35,  and  C  to  No.  69. 

1 10.  Make  a  diagram  of  the  following  winding  and  show  three 
collecting  rings   connected  to  conductors    I,  19  and   37.     The 
winding  is  a  four-pole,  two-circuit,  single  drum  winding  with  54 
conductors  connected  up  as  follows  :    1-14-27-40-53  .  .  .  3-16- 
29-42  and  back  to  i. 

in.  A  5o-kilowatt  direct-current  dynamo  is  to  be  used  as  an 
72-ring  converter.  What  is  its  capacity  rating  when  n  =  2,  when 
n  =  3,  when  n  =  4,  and  when  n  =  6  ?  Ans.  42.5,  66,  81,  and 
96  kilowatts. 

112.  A  three-ring  converter  is  to  supply  direct  current  at  500 
volts  to  a  street-car  line.  At  what  voltage  must  the  alternating 
currents  be  delivered  to  the  machine  ?  The  step-down  transfor- 
mation is  accomplished  by  three  similar  transformers  with  their 


THE   ROTARY    CONVERTER.  229 

primaries  A-connected  to  the  high  voltage  mains  and  their  sec- 
ondaries Y-connected  to  the  three  rings  of  the  converter.  What 
is  the  ratio  of  transformation  of  each  transformer,  the  line  voltage 

N" 
being  10,000  volts?     Ans.   306  volts,     -j^j-  =  0.0530. 

113.  A  two-ring  converter  with  negligible  armature  resistance 
and  reactance  takes  current  over  a  o.2-ohm  line  through  a  reac- 
tion coil   having  0.3  ohm  reactance.     The  effective   alternating 
electromotive  force  at  the  generator  terminals  is  kept  constant  at 
1 20  volts.     The  excitation  of  the  converter  can  be  varied  at  will. 
Find  greatest  value  of  alternating  electromotive  force  of  converter 
for  which  it  can  run  as  a  synchronous  motor,  and  find  the  corre- 
sponding value  of  the  direct  electromotive  force  of  the  converter. 

Find  the  greatest  direct-current  load  which  the  converter  can 
carry  without  dropping  out  of  synchronism  (a)  when  the  direct 
voltage  of  the  converter  is  at  its  maximum  and  (fr)  when  the 
direct  voltage  of  the  converter  is  5  volts,  I  o  volts,  1 5  volts,  20 
volts,  and  25  volts  respectively  less  than  the  maximum.  In  this 
problem  ignore  friction  and  other  losses  in  the  machine.  Ans. 
216.25  volts,  306  volts,  (a)  zero,  (b)  4.24  amperes,  7. 59  amperes, 
11.53  amperes,  15.32  amperes,  19.15  amperes. 

114.  Plot  the  curve  of  which  the  ordinates  represent  the  in- 
stantaneous values  of  the  current  in  an  armature  conductor  of  a 
four-pole,  three-ring  converter  with  multicircuit  winding,  /  being 
250  amperes  :  (a)  when  the  conductor  is  adjacent  to  the  connec- 
tion of  a  ring,  and  (&)  when  the  conductor  is  midway  between 
the  connections  of  two  rings. 

115.  A  loo-kilowatt  direct-current  generator  is  provided  with 
5  collector  rings.      Find  its  power  rating  as  a  rotary  converter, 
based  upon  average  armature  heating.     Ans.    1 8 1  kilowatts. 


CHAPTER   XIV. 


THE   INDUCTION   MOTOR. 

149,  The  induction  motor. — It  has  already  been  pointed  out 
that  the  successful  employment  of  alternating  current  for  motive 
purposes  depends  upon  the  use  of  the  induction  motor  driven  by 
polyphase  currents.  The  induction  motor  consists  of  a  primary 
member  and  a  secondary  member,  each  with  a  winding  of  wire. 
The  primary  member  is  usually  stationary,  and  is  often  called  the 
stator.  The  secondary  member  is  usually  the  rotating  member, 


Fig.  179. 


Fig.  180. 


and  is  often  called  the  rotor.  Fig.  179  shows  a  rotor  of  the 
squirrel-cage  type.  It  consists  of  a  drum  A  built  up  of  circular 
sheet-iron  disks  ;  near  the  periphery  of  this  drum  are  a  number 
of  holes  parallel  to  the  axis  of  the  drum  ;  in  these  holes  heavy 
copper  rods  b  are  placed,  and  the  projecting  ends  of  these  rods 
are  soldered  to  massive  copper  rings,  r,  one  at  each  end  of  the 
drum.  Another  type  of  rotor  is  described  later. 

230 


THE   INDUCTION   MOTOR. 


231 


The  stator  is  a  laminated  iron  ring,  FFt  Fig.  1 80,  closely  sur- 
rounding the  rotor.  This  ring  is  slotted  on  its  inner  face,  as 
shown,  windings  are  arranged  in  these  slots,  and  these  windings 
receive  currents  from  poly- 
phase supply  mains.  These 
polyphase  currents  produce 
in  the  stator  a  rotating  state 
of  magnetism,  the  action  of 
which  on  the  rotor  is  the  same 
as  the  action  of  an  ordinary 
field  magnet  in  rotation. 
Thus  Fig.  181  shows  a 
squirrel -cage  rotor  A  sur- 
rounded by  an  ordinary  field 
magnet  rotating  in  the  direc- 
tion of  the  curved  arrows. 

This  motion  of  the  field  magnet  induces  currents  in  the  short- 
circuited  copper  rods  of  the  rotor ;  the  field  magnet  exerts  a 
dragging  force  on  these  currents  and  causes  the  rotor  to  rotate. 

No  electrical  connec- 
tions of  any  kind  are 
made  to  the  rotor.  The 
next  article  describes 
the  stator  windings  and 
explains  the  manner  in 
which  these  windings 
produce  the  rotating 
state  of  magnetism  in 
the  stator. 

150.    Stator   windings 
and   their    action. — The 
stator  windings   are   ar- 
ranged in  the  slots  s,  Fig. 
1 80,  in  a  manner  exactly  similar  to  the  arrangement  of  the  wind- 


232 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


ings  of  the  two-phase  or  three-phase  alternator  armature,  accord- 
ing as  the  motor  is  to  be  supplied  with  two-  or  three-phase  currents. 
Fig.  182  shows  an  end  view  of  a  four-pole  two-phase  induc- 
tion motor.  In  this  figure  the  outline,  only,  of  the  rotor  is 
shown  ;  the  stator  conductors  are  represented  in  section  by  the 
small  circles  ;  the  slots  are  omitted  for  the  sake  of  clearness  ; 
and  the  end  connections  of  half  the  stator  conductors  are  shown 


Fig.  183. 

in  Fig.  183.  The  stator  conductors  are  arranged  in  two  distinct 
circuits.  One  of  these  circuits  includes  all  of  the  conductors 
marked  A  and  it  receives  current  from  one  phase  of  a  two-phase 
system  ;  the  other  circuit  includes  all  of  the  conductors  marked 
B  and  it  receives  current  from  the  other  phase  of  the  two-phase 
system.  The  terminals  of  the  B  circuit  are  shown  at  ttf ',  Fig.  183. 
The  conductors  which  constitute  one  circuit  are  so  connected 
that  the  current  flows  in  opposite  directions  in  adjacent  groups 


THE   INDUCTION   MOTOR. 


233 


of  conductors  as  indicated  by  the  arrows  in  Fig.  183.  The 
radial  lines  in  Fig.  183  represent  the  stator  conductors  and  the 
curved  lines  represent  the  end  connections,  as  in  the  winding 
diagrams,  Figs.  103  to  110. 

The  action  of  a  band  of  conductors  between  two  masses  of 
iron  is  shown  in  Figs.  184  and   185.     The  small  circles  in  these 


iron 


iron 


1OO0Q0! 


** y' 

irbrC  \ 

up     Howinjf  currertty 
Fig.  184. 


Cf6n~~ 

flowing  currenfy 
Fig.  185. 


figures  represent  the  conductors  in  section  ;  conductors  carrying 
down-flowing  currents  are  marked  with  crosses,  those  carrying 
up-flowing  currents  are  marked  with  dots,  and  those  carrying  no 
current  are  left  blank.  The  action  of  the  currents  in  these  bands 
of  conductors  is  to  produce  magnetic  flux  along  the  dotted  lines 
in  the  directions  of  the  arrows. 

The  lines  A'  and  B'  in  Figs.  186,  187  and   188  are  supposed 
to  rotate  and  their  projections  on  the  fixed  line  ef  represent  the 


Fig.   186. 


234 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


instantaneous  values  of  the  alternating  currents  in  the  A  and  B 
conductors  respectively. 


Fig.  187. 


Fig.  1 86  shows  the  state  of  affairs  when  the  current  in  conduc- 
tors A  is  a  maximum  and  the  current  in  conductors  B  is  zero. 
The  dotted  lines  indicate  the  trend  of  the  magnetic  flux.  This 


THE   INDUCTION   MOTOR.  235 

flux  enters  the  rotor  from  the  stator  at  the  points  marked  N  and 
leaves  the  rotor  at  the  points  marked  5. 

Fig.  187  shows  the  state  of  affairs,  |-  of  a  cycle  later,  when 
the  current  in  the  B  conductors  has  increased  and  the  current  in 
the  A  conductors  has  decreased  to  the  same  value.  The  points 
N  and  .S"  have  moved  over  y1^  of  the  circumference  of  the  stator 
ring. 

Fig.  1 88  shows  the  state  of  affairs,  after  another  eighth  of  a 
cycle,  when  the  current  in  the  B  conductors  has  reached  its 
maximum  value  and  the  current  in  the  A  conductors  has  dropped 
to  zero.  The  points  N  and  5  have  moved  again  over  -^  of  the 
circumference  of  the  stator  ring. 

This  motion  of  the  points  N  and  5  is  continuous,  and  these 
points  make  one  complete  revolution  (in  a  four-pole  motor)  dur- 
ing two  complete  revolutions  of  the  vectors  A'  and  Br  or  while 
the  alternating  currents  supplied  to  the  stator  windings  are  pass- 
ing through  two  cycles.  In  general 

-i 

in  which  n  is  the  revolutions  per  second  of  the  stator-magnetism, 
/  is  the  number  of  pairs  of  poles  N  and  5,  and/  is  the  frequency 
of  the  alternating  currents  supplied. 

Three-phase  stator  winding. — When  an  induction  motor  is 
driven  by  three-phase  currents,  the  stator  conductors  are  arranged 
in  three  distinct  circuits  A,  B  and  C,  which  are  either  A-  or 
Y-connected  to  the  supply  mains.  Fig.  1 89  shows  the  complete 
connections,  for  four  poles,  of  the  A  circuit  with  its  terminals  tt' . 
The  B  and  C  circuits  are  similarly  connected. 

In  general,  the  72-phase  stator  winding  for  2p  poles  has  2pn 
equidistant  bands  of  conductors.  The  ist,  (n  4-  i)th,  (2/2  -f  i)th, 
etc.,  bands  are  connected  in  one  circuit,  so  that  currents  flow  op- 
positely in  adjacent  bands,  and  this  circuit  takes  current  from  one 
phase  of  the  72-phase  system.  The  2d,  (;/  -f  2)th,  (2n  +  2)th, 
etc.,  bands  are  similarly  connected  in  another  circuit  and  take 


236 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


current  from  the  second  phase  of  the  ^-phase  system.     The  3d, 
(n  +  3)th,  (zn  -f  3)th,  etc.,    bands    are    similarly   connected    in 


B 


Fig.  189. 


another  circuit  and  take  current  from  the   third  phase  of  the 
//-phase  system  ;  and  so  on. 

151.  The  production  of  a  rotating  magnetic  field  by  means  of  two-phase  cur- 
rents without  an  iron  core. — A  coil  A  A,  Fig.  190,  with  its  center  at/  and  its  plane 
at  right  angles  to  a,  is  placed  inside  of  a  larger  coil  JBJB,  the  center  of  this  coil  being 
likewise  at/  and  its  plane  at  right  angles  to  b. 


Fig.  190. 


THE   INDUCTION    MOTOR.  237 

Let 

i  '  =  A  sin  ut 

be  the  alternating  current  flowing  in  coil  AA>  and 

i"^=B  cos  wt 

be  the  current  flowing  in  coil  BB.  The  magnetic  field,  a,  at  /  due  to  coil  A  A  is 
parallel  to  the  line  a  and  proportional  to  i/.  We  may,  therefore,  write 

a  =  F  sin  ut 
where  F  is  a  constant. 

The  magnetic  field  b  at/  due  to  coil  BB  is  parallel  to  the  line  b  and  proportional 
to  *",  so  that  if  the  number  of  turns  of  wire  in  coil  BB  is  properly  chosen  we  may 

write 

b  =  F  cos  ut 

The  resultant  field  at/  (equal  to  V a2  -f-  ^2)  is  constant  in  magnitude  and  it  rotates 
at  an  angular  velocity  u  ;  a  is  its  JF  component  and  b  is  its  y- component. 

152.  Preliminary  discussion  of  the  action  of  the  induction  motor. 

— The  complete  theoretical  discussion  of  the  action  of  the  induc- 
tion motor  is  given  later  and  is  in  many  respects  similar  to  the 
theory  of  the  transformer.  Many  important  details  of  the  action 
of  the  induction  motor,,  however,  are  most  easily  explained  by 
looking  upon  the  induction  motor  as  a  rotor  influenced  by  a  ro- 
tating field  magnet. 

Torque  and  speed. — Let  n  be  the  revolutions  per  second  of  the 
field  and  n'  the  revolutions  per  second  of  the  rotor.  When  n  =  n' 
the  rotor  and  field  turn  at  the  same  speed,  so  that  their  relative 
motion  is  zero  ;  no  electromotive  force  is  then  induced  in  the 
rotor  conductors  and  no  current,  and  therefore  the  rotating  field 
exerts  no  torque  upon  the  rotor.  As  the  speed  of  the  rotor  de- 
creases the  relative  speed  of  rotor  and  field  increases,  and  there- 
fore the  electromotive  force  induced  in  the  rotor  conductors,  the 
currents  in  the  conductors,  and  the  torque  with  which  the  field 
drags  the  rotor,  all  increase.  If  the  whole  of  the  field  flux  were 
to  pass  into  the  rotor  and  out  again  in  spite  of  the  demagnetizing 
action  of  the  currents  in  the  rotor  conductors,  then  the  torque 
would  increase  in  strict  proportion  to  n  —  nr ,  but  in  fact  a  larger 
and  larger  portion  of  the  field  flux  passes  through  the  space  be- 
tween stator  and  rotor  conductors  as  the  speed  of  the  rotor  de- 
creases and  this  magnetic  leakage  causes  the  torque  to  increase 


ELEMENTS  OF  ALTERNATING  CURRENTS. 


more  and  more  slowly  as  n  —  n*  increases,  in  some  cases  *  reach- 
ing a  maximum  value  and  then  decreasing  with  further  increase 
of  n  —  n' . 

Fig.  191  shows  the  typical  relation  between  torque  and  speed 
of  an  induction  motor.     Ordinates  of  the  curve  represent  torque 


Fig.  191. 


and  abscissas  measured  from  0  represent  rotor  speeds.  The 
rotor  is  said  to  run  above  synchronism  when  it  is  driven  so 
that  n'  >  n. 

Discussion  of  torque-speed  curve. — The  torque  acting  on  the 
rotor  is  proportional  to  the  product  of  the  rotor  current  i  into 
the  flux  <£>',  which  passes  into  the  rotor  from  the  rotating  field 
magnet.  Therefore  we  may  write 

T=  k<b'i  (i) 

The  field  flux  <I>  of  the  induction  motor  is  fixed  in  value  or 
constant,  and  the  difference  <I>  —  <£>'  is  proportional  to  i.  There- 
fore we  may  write  (<£>  —  <f>')  =  k'i,  or 

<£'  =  <£  -k'i  (ii) 

The  rotor  current  i  is  proportional  to  (n  —  n')  and  to  4>r. 
Therefore  we  may  write 

i  =  k"(n  -  n')  <&'  (iii) 

*  In  every  case,  if  one  makes  n  —  n'  large  enough  by  driving  the  rotor  backwards 
so  that  nf  becomes  negative. 


THE    INDUCTION    MOTOR.  239 

Eliminating  4>'  and  i  from  equation  (i)  with  the  help  of  equa- 
tions (ii)  and  (iii)  and  reducing,  we  have 

a(n  —  nf) 
=  l  +  6(n-n'Y 

in  which  a  and  b  are  constants  depending  upon  <l>,  k,  k'  and  k"  . 
This  equation  is  plotted  in  Fig.  191,  in  which  the  abscissas  rep- 
resent speed  of  rotor  n'  .  The  curve  crosses  the  jr-axis,  for 
«'  =  ». 

Use  of  starting  resistance  in  the  rotor  windings.  —  The  speed  of 
rotor  for  which  the  maximum  torque  occurs  depends  upon  the 
resistance  of  the  rotor  windings,  and  it  is  advantageous  to  provide 
at  starting  such  resistance  in  these  windings  as  to  produce  the 
maximum  torque  at  once,  this  resistance  being  cut  out  as  the 
motor  approaches  full  speed. 

Efficiency  and  speed.  —  Let  T  be  the  torque  with  which  the  ro- 
tating field  drags  on  the  rotor  ;  then  2irn'  T  is  the  power  taken 
up  by  the  rotor  to  be  given  off  its  belt  pulley.  Also  T  is  the 
reacting  torque  which  opposes  the  rotation  of  the  field,  so  that, 
ignoring  friction,  2TrnT  is  the  power  required  to  maintain  the 
rotation  of  the  field.  Therefore,  ignoring  friction  losses,  2TrnT  is 
the  power  intake  and  27rn'  T  is  the  power  output  of  the  motor, 
so  that 

T      nr 


is  the  efficiency  of  the  machine.  This  equation  shows  that  the 
efficiency  of  an  induction  motor  is  zero  when  the  rotor  stands 
still,  that  it  increases  as  the  rotor  speeds  up  and  approaches  ioo$& 
(ignoring  field  losses  and  friction)  as  the  rotor  speed  approaches 

n' 
the  field  speed.     The  ratio—  ranges  from  .85  to  .95  or  more  in 

commercial  induction  motors  under  full  load. 

Efficiency  and  rotor  resistance.  —  For  a  given  difference  n  —  nf 
between  field  speed  and  rotor  speed,  given  electromotive  force  is 
induced  in  the  rotor  conductors,  and  the  less  the  rotor  resistance 


240  ELEMENTS   OF   ALTERNATING   CURRENTS. 

the  greater  the  current  produced  by  this  electromotive  force  and 
the  greater  the  torque.  Therefore  a  given  induction  motor  will 
develop  its  full  load  torque  for  a  small  value  of  n  —  n'  or  for  a 

nr 
large  value  of  —  (efficiency)  if  its  rotor  resistance  is  small.      High 

efficiency  depends,  therefore,  upon  low  rotor  resistance. 

153.  The  induction  generator.  —  When  the  rotor  of  an  induction 
motor  is  driven  above  synchronism  (nf  >  n\  by  an  engine  for 
example,  the  torque  is  reversed  and  opposes  the  motion  of  the 
rotor  so  that  27rnfTis  input  and  2irnT  is  output.     That  is,  the 
machine  takes  power  from  the  engine  to  drive  its  rotor  and  gives 
out  power  from  its  stator  windings.     This  output  of  power  is  in 
the  form  of  polyphase  currents  the  frequency  of  which  is  fixed  by 
the  frequency  of  the  alternator  (or  synchronous  motor)  which  is 
connected  to  the-  stator  windings. 

154.  The  driving  of  induction  motors  by  single-phase  alternat- 
ing current.  —  This  is  accomplished  by  connecting  the  two  stator 
circuits  A  and  B  (case  of  two-phase  motor)  in  parallel  to   the 
single-phase  supply  mains  at  the  same  time  connecting  in  series 
with  A  a  resistance  R  (see  Fig.  192).     The  currents  in  the  cir- 

cuits  A  and  B  then  differ  in  phase 
on  account  of  the  dissimilarity  of 
the  circuits,  and  the  motor  starts. 
When  the  motor  is  well  under  way 
one  of  the  windings  A  or  B  may  be 
open-circuited^  the  other,  only,  being 
left  connected  to  the  mains.  A  two- 


Fig-  192-  phase,  or  even  a  three-phase  motor 

operates  fairly  well  under  these  conditions,  except  that  excessive 
current  is  required  at  starting  to  give  a  good  starting  torque. 
The  resistance  R  may  be  replaced  with  advantage  by  a  condenser, 
especially  in  case  of  a  small  motor. 

155.  The  action  of  the  polyphase  alternator  as   an  induction 
motor  when  being  started  as  a  synchronous  motor.  —  The  winding 


THE    INDUCTION   MOTOR. 


24I 


of  the  polyphase  armature  is  identical  to  the  stator  or  primary 
winding  of  an  induction  motor.  When,  at  starting,  the  armature 
is  connected  to  the  polyphase  supply  mains  a  rotating  state  of 
magnetism  is  set  up  in  the  armature  core.  This  rotating  mag- 
netism exerts  a  dragging  torque  on  the  field  magnet,  especially  if 
the  field  coils  are  short-circuited,  and  the  reacting  torque  of  the 
field  upon  the  armature  sets  the  latter  rotating  in  a  direction  op- 
posite to  the  direction  of  its  rotating  magnetism. 

156.  The  induction  wattmeter  is  essentially  a  two-phase  induc- 
tion motor,  of  which  the  driving  torque  is  proportional  to  the 
watts  delivered  to  the  re- 
ceiving circuit.  The  ar- 
mature disk  is  large  and 
one  edge  of  it  moves  be- 
tween the  poles  of  a  per- 
manent steel  magnet  (not 
shown  in  the  figure)  which 
causes  the  speed  to  be 
proportional  to  the  driving 
torque  as  in  a  Thompson 
wattmeter. 

The  total  current  de- 
livered to  the  receiving 
circuit  passes  through  the 
coil  of  coarse  wire  on  the 
lug  A  of  the  laminated  iron 
core,  Fig.  193.  The  lugs 
BB  are  wound  with  fine  wire  which  is  connected  across  the  mains. 

The  alternating  current  /  in  coil  A  produces  a  magnetic  flux 
in  lug  A,  which  is  proportional  to  and  in  phase  with  /.  This  flux 
induces  electromotive  force  in  the  disk  DD,  which  is  90°  ahead 
of  the  flux  in  phase,  and  this  electromotive  force  produces  cur- 
rent in  the  disk,  which  is  in  phase  with  it.  Therefore  the  eddy 
current  in  the  disk  is  proportional  to  and  90°  ahead  of  /.  This 


Fig.  193. 


242  ELEMENTS   OF   ALTERNATING    CURRENTS. 

eddy  current  flows  along  the  circular  lines  shown  in  the  figure, 
passing  under  the  ends  of  the  lugs  BB. 

The  electromotive  force  E  between  the  mains  produces  in  coil 
BB  a  current  which  is  proportional  to  and  nearly  90°  behind  E. 
This  current  in  its  turn  produces  through  B  and  B  a  flux  which 
is  90°  behind  and  proportional  to  E.  The  phase  difference  be- 
tween this  flux  and  the  eddy  current  in  the  disk  is  equal  to  the 
phase  difference  0  between  E  and  /. 

The  eddy  current  in  passing  under  the  lugs  BB  is  pushed  side- 
ways by  the  flux  from  BB  with  a  force  the  average  value  of 
which  is  proportional  to  the  product  of  maximum  eddy  current 
X  maximum  flux  x  cosine  of  phase  difference  between  the  two, 
which,  according  to  the  statements  above,  is  proportional  to  El 
cos  6,  or  to  the  power  delivered  to  the  receiving  circuit.  The 
speed  of  the  disk  is  therefore  proportional  to  power  delivered  and 
the  total  revolutions  of  the  disk  in  a  given  time  is  proportional 
to  the  total  work  delivered. 

GENERAL  THEORY  OF  THE  INDUCTION  MOTOR. 
157.  The  general  alternating-current  transformer. — The  general 
theory  of  the  induction  motor  is  best  developed  by  considering 
at  once  the  most  general  type  of  machine,  a  multipolar  multi- 
phase motor,  of  which  the  rotor  is  wound  in  precisely  the  same 
way  as  the  stator,  the  rotor  windings  being  connected  to  collect- 
ing rings,  so  that"  the  currents  induced  in  the  motor  windings 
may  be  available  for  outside  purposes.  Such  a  machine  we  will 
call  the  general  alternating-current  transformer.  Thus,  a  2p- 
pole,  ^-phase  machine  would  have  its  stator  conductors  arranged 
in  q  distinct  circuits,  each  taking  current  from  one  phase  of  a  q- 
phase  system  ;  furthermore,  each  circuit  would  include  2p  equi- 
distant groups  of  conductors  so  connected  that  a  current  in  that 
circuit  would  flow  in  opposite  directions  in  adjacent  groups.  The 
rotor  conductors  would  be  similarly  arranged  in  q  *  distinct  cir- 

*  Stator  and  rotor  are  not  necessarily  wound  for  the  same  number  of  phases,  but 
the  discussion  is  simplified  by  such  an  arrangement. 


THE   INDUCTION    MOTOR. 


243 


cuits,  each  connected  to  a  pair  of  collecting  rings  and  supplying 
current  to  an  outside  receiving  circuit. 

Of  course,  no  such  induction  motors  *  are  ever  actually  built, 
but  it  is  important  to  have  clearly  in  mind  the  details  of  the 
machine  to  which  the  following  discussion  applies. 

Fig.  194  shows  a  little  more  than  one-sixth  part  of  the  cir- 
cumference of  a  six-pole,  three-phase  machine.  The  three 
groups  of  stator  conductors,  A,  B  and  C,  belong  one  to  each  of 
the  three  circuits  formed  by  the  stator  windings,  and  the  three 
groups  of  rotor  conductors,  A',  B' 
and  C1 ',  belong  one  to  each  of  the 
three  circuits  formed  by  the  rotor 
windings. 

When  the  rotor  of  such  a  machine 
is  stationary  the  machine  acts  simply 
as  a  transformer  taking  ^-phase  cur- 
rents from  the  supply  mains  into  its 
stator  windings  and  .giving  out  q- 
phase  currents  of  the  same  frequency 
from  its  rotor  windings. 

Remark.  —  The  immediately  fol- 
lowing discussion  is  based  upon  the 
ideal  induction  motor,  of  which  the 
primary  and  secondary  windings  have 
no  resistance  ;  the  magnetic  flux 
through  the  stator  windings  all  passes 
into  the  rotor,  and  the  magnetizing  current  is  negligible. 
Throughout  the  discussion  the  stator  and  rotor  are  supposed  to 

*  Steinmetz  has  proposed  the  use  of  such  induction  motors  for  street  railway  work. 
Two  similar  motors  are  used  on  each  car,  one  geared  to  each  axle.  At  starting  and 
for  slow  running,  motor  No.  I  takes  currents  into  its  stator  windings  from  polyphase 
trolley  wires,  and  supplies  polyphase  currents  from  its  rotor  windings  to  the  stator  of 
motor  No.  2,  the  starting  resistance  being  connected  in  the  rotor  circuits  of  motor 
No.  2.  With  such  an  arrangement  the  limit  of  speed  is  one-half  of  synchronous 
speed  (n'  =  y£n'),  and  the  efficiency  at  given  speed  is  doubled.  For  fast  running 
both  motors  take  current  directly  from  the  trolley  wires. 


Fig.  194. 


244  ELEMENTS   OF   ALTERNATING   CURRENTS. 

be  wound  with  the  same  number  of  conductors  and  for  the  same 
number  of  phases. 

158.  Rotor  electromotive  forces  referred  to  stator, — The  electro- 
motive forces  induced  in  the  rotor  conductors,  and  also  the 
electromotive  forces  induced  in  the  stator  conductors,  may  be 
ascribed  to  the  rotating  stator  magnetism.  Let  n  be  the  speed 
of  the  stator  magnetism,  and  nf  the  speed  of  the  rotor.  Then 
the  speed  of  the  stator  magnetism  referred  to  stator  conductors 
is  ;/,  and  the  speed  of  the  stator  magnetism  referred  to  rotor  con- 
ductors is  n  —  n' .  Consider  (a)  the  varying  electromotive  force 
which  is  induced  in  a  given  stator  conductor,  and  (b]  the  instan- 
taneous values  of  electromotive  force  induced  in  the  various  rotor 
conductors  as  they  pass  the  given  stator  conductor.  These  two 
electromotive  forces  are  at  each  instant  in  the  ratio  of  n  to  n  —  n' , 
because  they  are  produced  by  the  same  lines  of  force  sweeping 
past  the  stator  conductor  at  speed  n  and  sweeping  past  the  suc- 
cessive rotor  conductors  at  speed  n  —  n' .  Therefore  the  two 
electromotive  forces  (a)  and  (b}  above  are  of  the  same  frequency, 
their  ratio  is  n  to  n  —  n' ,  and  they  are  in  phase  with  each  other. 
Of  course  the  electromotive  force  induced  in  one  of  the  circuits 
of  the  stator  winding  is  equal  and  opposite  to  the  impressed 
electromotive  force  E'  which  acts  on  that  circuit. 

Rotor  electromotive  forces  referred  to  rotor. — When  the  rotor  is 
at  a  standstill  the  relative  speed  of  rotor  and  stator-magnetism  is 
n,  and  the  electromotive  force  induced  in  a  given  rotor  conductor 
is  equal  to  the  electromotive  force  E  induced  in  a  stator  con- 
ductor and  of  the  same  frequency  /.  When  the  rotor  runs  at 
speed  n'  the  relative  speed  of  rotor  and  stator  magnetism  drops 
to  n  —  n1 ',  and  the  electromotive  force  induced  in  a  given  rotor 
conductor  is  decreased  in  the  ratio  n  to  n  —  n'  both  in  value 
and  in  frequency,  its  value  becoming  sE  and  its  frequency  sf 
where 

n  —  n' 
*=-  (113) 


THE   INDUCTION   MOTOR.  245 

This  quantity,  s,  is  much  used  in  the  theory  of  the  induction 
motor,  and  is  called  the  slip. 

159.  Rotor  currents  referred  to  stator. — When  the  secondary 
or  rotor  circuits  are  open,  the  primary  or  stator  current  m  in 
each  primary  circuit  is  called  the  magnetizing  current.     Let  /" 
be  the  current  in  each   rotor  circuit  when  the  rotor  circuits  are 
closed,  and  let  I'  be  the  additional  current,  over  and  above  m, 
which  flows  in  each  stator  circuit.     Then,  as  in  the  case  of  the 
transformer,  the  magnetizing  action  of  I"  is  balanced  at  each  in- 
stant by  the  magnetizing  action  of  I' .     In  the  following  discussion 
the  currents  ;;/  are  not  considered. 

Consider  (a)  the  current  in  a  given  stator  conductor  and  (b) 
the  instantaneous  values  of  current  in  the  various  rotor  conduc- 
tors as  they  pass  the  given  stator  conductor.  These  two  currents 
are  at  each  instant  equal  and  opposite,  otherwise  the  magnetiz- 
ing action  of  the  one  could  not  be  at  each  instant  balanced  by  the 
magnetizing  action  of  the  other.  Therefore  the  two  currents  (a) 
and  (b)  above  are  of  the  same  frequency,  they  are  equal  to  each 
other,  and  they  are  opposite  to  each  other  in  phase. 

Rotor  currents  referred  to  rotor. — When  the  rotor  is  at  stand- 
still, the  current  in  a  given  rotor  conductor  has  the  same  value  7 
and  the  same  frequency  f  as  the  current  in  a  stator  conductor. 
When  the  rotor  speed  is  n'  the  current  in  the  given  rotor  con- 
ductor remains  equal  in  value  to  the  current  in  a  stator  conduc- 
tor, but  it  drops  in  frequency  in  the  ratio  of  (n  —  n')  -r-  n,  that  is, 
its  frequency  becomes  sf. 

Remark.— When  one  is  studying  the  mutual  action  of  stator 
and  rotor  it  is  convenient  to  refer  rotor  electromotive  forces  and 
rotor  currents  to  the  stator. 

160.  The  vector  diagram  of  the  induction  motor. — Let  £v  Fig. 
195,  represent  the  electromotive  force  acting  upon  one  circuit  of 
the  stator  winding.     The  secondary  electromotive  force  E*  per 

*  The  precise  meaning  of  E2  is  most  clearly  stated  when  there  are  ?pq  conductors 
on  stator  and  rotor,  one  for  each  band  of  actual  conductors,  so  that  2/  conductors  con- 


246 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


A 

\ 


E 


Fig.  195. 


circuit  is  equal  to  sEl  and  opposite  to  E^  in  phase.  This  elec- 
tromotive force  £2  has  a  frequency  sf  referred  to  the  outside  re- 
ceiving circuit  to  which  it  supplies  current. 
Let  r  be  the  resistance  of  this  circuit,  and  let  L 
be  its  inductance,  or  sx(==  2nrsfL)  its  reactance. 
Then  /2  is  equal  to  £2  H-  (r+j-  sx)  and  tan  0 
=  sxjr.  The  current  7:  in  the  stator  circuit  is 
equal  and  opposite  to  /2. 

The  input  of  power  per  phase,  that  is,  the 
power  delivered  to  each  stator  circuit,  is  EJ^ 
cos  6  ;  and  the  electrical  output  of  power  per 
phase,  that  is,  the  output  from  each  rotor  cir- 
cuit, is  E2I2  cos  6  or  sE^  cos  0,  since  £2  =  sE^ 
and  /j  =  /2.  The  difference  between  electrical 
input  and  electrical  output  is  mechanical  out- 
put, or  power  delivered  by  belt.  This  is  (1  —  ^)^171  cos  0,  or 
(1  —  s)  I  s  times  the  electrical  output. 

Example. — A  four-pole,  three-phase  machine  takes  current  at 
220  volts  and  60  cycles  per  second.  When  the  rotor  is  station- 
ary it  gives  out  three-phase  currents  at  full  frequency,  the  amount 
of  output  depending,  of  course,  upon  the  resistance  and  reactance 
of  the  receiving  circuits.  The  speed  of  the  stator  magnetism  is 
30  revolutions  per  second.  When  rotor  speed  is  20  revolutions  s 
[  =  (n  —  n')/n]  is  equal  to  ^,  so  that  the  rotor  delivers  three- 
phase  currents  at  73  %  volts  and  at  20  cycles  per  second,  and  of 
the  total  power  delivered  to  the  machine  %  is  electrical  output  and 
y±  is  mechanical  output. 

161.  The  actual  induction  motor. — The  behavior  of  the  actual 
induction  motor  deviates  from  the  above  described  ideal  action, 
because  of  the  resistance  of  the  stator  windings,  because  of  eddy 
current  and  hysteresis  losses  in  the  iron  and  because  of  magnetic 

stitute  one  circuit.  Then  £l  is  the  electromotive  force  in  a  given  stator  circuit  and  E2 
is  the  electromotive  force  in  the  successive  rotor  circuits  as  they  pass  by  the  given 
stator  circuit.  E2  is  opposite  to  £v  equal  to  sEv  and  referred  to  the  stator  its  fre- 
quency is  the  same  as  the  frequency  of  £r 


THE   INDUCTION   MOTOR.  247 

leakage.  The  effect  of  each  of  these  things  is  rather  small,*  there- 
fore their  mutual  influences  may  be  neglected.  The  effect  of  each 
will  consequently  be  considered  by  itself.  The  effects  will  first 
be  discussed  in  a  general  way  with  the  help  of  the  vector  dia- 
gram, after  which  the  general  complex  equations  of  the  induction 
motor  will  be  established  and  Steinmetz's  solution  outlined.  The 
discussion  will  be  directed  mainly  to  the  mechanical  behavior  of 
the  machine,  that  is,  to  the  relation  between  torque  and  speed. 

The  effect  of  magnetic  leakage  is  equivalent  to  an  outside  in- 
ductance connected  in  series  with  the  stator  circuit.  The  rotor 
windings  may  be  considered  as  non-inductive. 

162.  Effect  of  magnetic  reluctance,  eddy  currents  and  hysteresis 
upon   the   action  of  an  induction   motor. — The  ideal    induction 
motor  takes  no  current  from  the  mains  into  its  stator  windings 
when  the  motor  is  running  at  synchronous  speed  (/'  =/).     The 
actual  induction  motor  running  in  synchronism  takes  sufficient 
current  to  overcome  the  magnetic  reluctance  of  the  iron  (stator 
and  rotor) ;  and  an  amount  of  power  equal  to  the  hysteresis  and 
eddy  current   loss  in  the  stator  iron  only,  inasmuch  as  the  mag- 
netic state  of  the  rotor  is  constant,  since  it  rotates  with  the  stator 
magnetism.     When  the  actual  induction  motor  is  running  at  any 
given  speed  it  takes  from  the  mains  the  above  current  and  power 
in  excess  of  what  an  ideal  motor  would  take  at  same  speed. 
Further,  there  is  eddy  current  and  hysteresis  loss  in  the  rotor 
iron  when  it  runs  below  synchronism  and  the  effect  of  this  loss 
is  to  slightly  increase  the  torque. 

163.  Effect  of  stator  resistance  upon  the  action  of  an  induction 
motor. — When  the  rotor  is  running  nearly  in  synchronism  with 
the  stator  magnetism  the  currents  in  the  stator  windings  are  very 
small  and  no  perceptible   portion   of  the  supply  electromotive 
force  is  needed  to  overcome  the  stator  resistance.     As  the  rotor 
is  slowed  up  the  stator  currents  increase  and  a  larger  and  larger 
portion  of  the  supply  electromotive  force  is  needed  to  overcome 

*  Not,  however,  so  small  as  in  the  simple  transformer. 


248 


ELEMENTS    OF   ALTERNATING   CURRENTS. 


the  stator  resistance.  The  result  is  that  the  core  flux  falls  off* 
slightly,  and  also  the  torque  acting  upon  the  rotor  falls  short  of 
its  ideal  value,  inasmuch  as  this  torque  depends  upon  both  flux 
and  rotor  currents. 

164.  Effect  of  magnetic  leakage  upon  the  action  of  an  induction 
motor.— As  in  case  of  the  simple  transformer  the  effect  of  mag- 
netic leakage  is  the  same  as  the  effect  of  an  outside  inductance 

connected  in  series  with  the  primary  (stator) 
windings,  a  separate  inductance  for  each  stator 
circuit.  Let  P  be  the  value  of  each  induc- 
tance and  let  x  (  =  a>P)  be  its  reactance  value. 
The  diagram  of  Fig.  196  represents  the 
action  of  an  induction  motor,  in  so  far  as  it 
is  affected  by  magnetic  leakage ;  A  and  £2 
(=  sA)  are  the  electromotive  forces  induced 
in  stator  and  rotor  windings  respectively  by 
the  magnetic  flux  which  passes  through  both. 
The  current  /2  is  determined  by  the  resistance 
and  reactance  f  of  the  secondary  circuit. 
The  primary  current  7j  is  equal  and  opposite 
to  /2.  The  line  xlv  at  right  angles  to  7p  rep- 
resents that  part  of  the  total  primary  elec- 
tromotive force  which  is  used  to  overcome  the  leakage  induc- 
tance P  or  to  balance  the  electromotive  force  induced  in  each 
primary  circuit  by  the  leakage  flux. 

165.  Calculation  of  leakage  reactance. — The  leakage  reactance 
x  per  circuit  is  equal  to  a>P(=  27rfP)  where  Pis  the  leakage  in- 
ductance per  circuit.     This  leakage  inductance  is  calculated,  as 
in  case  of  the  simple  transformer,  by  equation  (75),  namely 

*  Inasmuch  as  the  portion  of  the  supply  electromotive  force  w^iich  is  balanced  by 
the  induced  electromotive  force  in  the  stator  windings  is  decreased,  and,  therefore 
the  harmonically  varying  flux  which  induces  this  electromotive  force  must  decrease 
exactly  as  in  the  simple  transformer. 

f  Under  practical  conditions  the  rotor  circuits  are  non-inductive  and  the  angle  0r 
Fig.  196,  is  zero,  as  in  case  of  the  simple  transformer  feeding  a  non-inductive  receiv- 
ing circuit. 


Fig.  196. 


THE   INDUCTION    MOTOR.  249 

\.(X       Y 


I        \  ,  -   „   ^<s  / 

This  equation  gives  P  in  centimeters,  all  dimensions  being  in 
centimeters.  In  this  equation  \  is  the  length,  parallel  to  the 
shaft,  of  the  rotor  or  stator ;  /  is  the  sum  of  the  widths  of  all 
the  slots  in  which  the  windings  of  one  stator  circuit  are  wound  ; 
X  is  the  depth  of  the  stator  slots  ;  Y  is  the  depth  of  the  rotor 
slots,  and  g  is  the  clearance  space  between  stator  and  rotor. 
This  equation  assumes  that  stator  and  rotor  slots  are  of  the 
same  width,  that  they  are  wound  full  of  wire,  and  that  the  per- 
meability of  the  iron  lugs  between  the  slots  is  very  great,  so  that 
reluctance  of  iron  is  negligible. 

166.  Formulation  of  the  complex  equations  of  the   induction 
motor. — The  following  discussion  is  taken  from  Steinmetz  *  with 
the  same  changes  as  are  mentioned  in  Article   120.     Stator  and 
rotor  are  supposed  to  be  wound  for  the  same  number  of  phases. 
Let  N'  =  number  of  stator  conductors  per  phase. 
N"  =  number  of  rotor  conductors  per  phase. 
N' 


r^     =  resistance  of  stator  per  circuit. 

r2  —  resistance  of  rotor  per  circuit,  including  outside  re- 
sistance (non-inductive). 

x  =  reactance  value  coP  of  primary  leakage  inductance 
per  phase. 

Z    =  ri  +JX- 

Ef   =  primary  impressed  electromotive  force  per  phase. 

A  =  that  part  of  E'  which  is  used  to  balance  the  electro- 
motive force  induced  in  each  stator  circuit  by  the 
magnetic  flux  which  passes  through  the  rotor. 

B     =  secondary  induced  electromotive  force  per  phase. 

/'    =  total  primary  current  per  circuit. 

M   =  magnetizing  current  per  circuit. 

I"    =  secondary  current  per  circuit. 

*«'  Alternating  Current  Phenomena,"  third  edition,  p.  221. 


250  ELEMENTS   OF   ALTERNATING   CURRENTS. 

M 

Yl  =  gl  —  jbl  —  —  j-  =  admittance  of  motor  per  phase  at 

zero  load,  that  is,  at  synchronous  speed. 

n  —  n1 
s    =  -  ,  where  n  is  the  speed  of  stator  magnetism 

and  //  '  is  speed  of  rotor. 

Remark.  —  In  calculating  gl  and  bl  use  equations  (71)  and  (72), 
in  which  We  and  Wh  are  eddy  current  loss  and  hysteresis  loss  in 
stator  iron  only,  and  G  is  magnetic  reluctance  of  stator  iron,  air 
gap,  and  rotor  iron.  For  calculations  near  standstill  it  would  be 
more  accurate  to  use  for  Wt  and  Wh  the  losses  in  both  stator  and 
rotor. 

The  ratio  of  A  to  B  is  a  when  the  rotor  stands  still,  and  a/s 
when  the  rotor  speed  is  n'  .  Further,  A  and  B  are  opposite  in 

phase,  so  that  aB 

A=  --  IT  (0 

The  secondary  current  is 

/"  -  *  (ii) 

'* 

The  part  of  I'  which  corresponds  to  I"  is  equal  to  —  /"/a  or 
B 


—  >  which,  added  to  M(=  Y^A)  gives  the  total  primary  cur- 
ar2 

rent.     That  is,  B  (iii) 

I'  =  Y.A  -- 
ara 

The  electromotive  force  used  to  overcome  the  impedance  of 
the  stator  windings  per  circuit  is  ZI1  ,  which,  added  to  A,  gives 
E'.  That  is, 

E'  =  A  +  ZI  '  (iv) 

With  the  help  of  equation  (i),  /'  and  E'  may  be  expressed  in 
terms  of  B,  giving 


THE   INDUCTION    MOTOR.  251 


It  is  desirable  to  express  I'  and  I"  in  terms  of  Er  as  follows  : 

Ef 


I"  =  - 


£\ 

*        J 


/'  =  +  A^  i_±_^_  (ng) 

I i  1 

s       ar2          s 

These  complex  equations  are  most  easily  handled  by  taking 
E'  as  the  reference  axis.  Then  E'  becomes  a  simple  quantity, 
and  the  components  of  /'  and  I"  are  easily  found  by  separating 
the  real  and  imaginary  parts  of  (n/)  and  (i  18)  respectively. 

When  the  components  of  I"  are  thus  found,  the  product  of 
r2  into  the  sum  of  the  squares  of  the  components  of  I"  gives  the 
electrical  output  of  the  rotor  per  phase,  and  the  product  of  this 

electrical  output  per  phase  by ,  as  stated  in  Article  1 59,  gives 

the  mechanical  output  per  phase. 

It  is  convenient  to  express  torque  in  terms  of  the  power  which 
the  torque  would  develop  at  synchronous  speed  ;/.  This  is  some- 
times called  "  synchronous  watts  "  for  brevity.  It  exceeds  the 
actual  mechanical  output  in  the  ratio  of  n  to  nf .  But  nln'  is  equal 

to »  therefore  mechanical  output  multiplied  by^ >  gives 

torque  expressed  in  synchronous  watts. 
Proceeding  as  above  we  find 

Component  of  I"  parallel  to  Ef  = ^~ 

u*  +  v* 

Component  of  /"  perpendicular  to  E'  =  -f-  ~2 — —^ 
Numerical  value  of  /"  = 


252  ELEMENTS   OF   ALTERNATING   CURRENTS. 

in  which 

u  =  a\  +  sr2  +  a\  (g^  +  bjc) 


a?s2r  E'Z 
Therefore  the  electrical  output  per  phase  is  -  ;  —  ^-  =-,  which 

U    -f  IT 

multiplied  by  -     -  gives  the  mechanical  output  per  phase,  and 

this  multiplied  by  the  number  of  phases  q  gives  the  total  me- 
chanical output  P,  for  which  the  expression  is 


in  which  c  is  written  for  a 

d  is  written  for 
aV22(l  +  2glrl  +  2b 

h  is  written  for  2aV2  (rl 
and 

y^  is  written  for  r*  +  x* 

Multiplying  P  by  ^-  —  gives  the  torque  2"  in  terms  of  syn- 

chronous watts.     Therefore 

scE'2 


The  ordinates  of  the  curves  in  Fig.  197*  represent  the  values 
of  P  and  of  T  (in  kilogram-meters)  and   the  abscissas  represent 

both   slip  s  I  =  —       -  j  and  speed  in  per  cent,  of  synchronism 

(  —  r     The  curves  are  calculated  for  a  three-phase,  eight-pole, 

2O-horse-power  (rated)  induction  motor  Y-connected  to  I  lo-volt 
6o-cycle  mains,  so  that  E'  =  63.6  volts.  The  rotor  winding  has 
been  reduced  to  an  equivalent  winding  for  which  a  —  1  : 

ampere 
^L  ~~  volts 

*  Taken  from  Steinmetz,  "Alternating  Current  Phenomena."     Calculations  verified 
by  P.  L.  Anderson  and  L.  A.  Freudenberger. 


THE   INDUCTION   MOTOR. 


253 


i 

-2 

-1 

( 

f 

1 

z 

8/Jje 

i 

E 

i 

§- 

-< 
+go- 

f\ 

* 

/ 

h70- 

-*0t*^ 

/ 

^"\ii 

Q 

X 

/  \|1 

-10- 

-zo 

-50 
-*0 

^.^^^^ 

1 

\ 

& 
~J 

oe_ 

- 

-20 

-40 

•powe* 

i 

^\ 

oweT" 

r 

y 

1  1  / 

-IQQ 

-100 

0 

^ 

10 

200 

500  %     Speed 

Fig.  197. 

A           Ampere 

fj  =  0.03  ohm 
x=  0.175  ohm 
r2  =  0.045  onm 

The  ordinates  of  the  curves  in  Fig.  198  *  represent  speed, 
efficiency,  power  factor,  torque  and  primary  current  for  various 
values  of  power  output.  The  curves  are  all  tangent  to  the  vertical 
line  of  which  the  abscissa  represents  the  maximum  power  output. 
These  curves  are  calculated  for  a  three-phase,  eight-pole  induc- 
tion motor  of  about  6-horse-power  rating,  A-connected  to  1  10- 
volt  mains  so  that  E'  is  equal  to  1  10  volts. 

ampere 


*  Taken  from  Steinmetz,  "  Alternating  Current  Phenomena.  '  '    Calculations  verified 
by  P.  L.  Anderson  and  L.  A.  Freudenberger. 


254 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


8000 


6000 


4000 


6000 


Fig.  198. 


=  O. 


ampere 
volts 

r^  =  o.  I  ohm 
x=  0.6  ohm 
r2  =  o.  i  ohm 

A  mere  indication  of  the  steps  by  which  the  curves  of  Fig. 
198  were  calculated  must  suffice.  Choose  a  series  of  values  of  s. 
For  each  of  these  calculate  mechanical  power,  speed  and  torque, 


THE   INDUCTION    MOTOR.  255 

from  equations  (119)  and  (120).  Separate  the  components  of  /' 
referred  to  E' ',  equation  (118),  and  calculate  the  values  of  these 
components  for  each  of  the  chosen  values  of  s.  From  these 
components  the  numerical  value  of  /',  the  power  factor,  the 
power  intake  and  the  efficiency  of  the  motor,  may  be  readily  cal- 
culated. The  power  factor  is  the  cosine  of  the  angle  between  E' 
and  /'  and  the  tangent  of  this  angle  is  equal  to  the  ratio  of  the 
two  components  of  /'. 

167,  Maximum  torque. — In  the  following  discussion  the  mag- 
netizing current  is  ignored  for  the  sake  of  simplicity,  so  that 
gl  =  b^  =  o.  Also  the  stator  and  rotor  conductors  are  supposed 
to  be  equal  in  number,  so  that  a==  1.  In  this  case  the  expres- 
sion for  torque,  equation  (120),  becomes 


The  value  of  slip  s  which  gives  maximum  T  is  found  by  the  con- 

J'T* 

dition  —-.—  =  o,  which  gives  as  the  value  of  s  for  maximum  torque 

,-±-     £=  (122) 


'  i    i^  **" 
Substituting  this  value  of  s  in  the  expression  for  T,  we  have 


(123) 


The  larger  value  of  T  is  the  negative  value  and  it  occurs  when 
rotor  speed  is  above  synchronism,  as  shown  in  Fig.  191. 

168.  Starting  torque. — At  starting  s—\,  which,  substituted  in 
equation  (121),  gives  the  value  of  the  starting  torque  TQ 

?  +  x* 

Now  the  value  of  maximum  torque  is,  according  to  equation 
(123),  independent  of  the  value  of  rv  but  variations  of  the  value 


256 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


of  r2  make  this  maximum  torque  occur  for  different  values  of  s, 
according  to  equation  (122).  Therefore  maximum  torque  occurs 
at  starting  when  the  value  of  r2  is  such  as  to  give  s  =  1  in  equa- 
tion (122). 

Therefore  to  give  maximum  torque  at  starting  we  must  have 


r2  —  <*r*  -f-  x* 

or  rotor  resistance  per  circuit  must  be  equal  to  primary  impe- 
dance per  circuit.     The  value  of  maximum  torque  at  starting  is 


T 


169.  The  graphical  solution  of  the  transformer  and  of  the  in- 
duction motor. — Consider  a  transformer  of  which  the  secondary 
delivers  current  to  a  non-inductive  receiving  circuit  of  variable 
resistance  R  " .  The  primary  current  is  that  which  would  be  de- 
livered to  a  circuit  of  inductance  P  and  of  which  the  resistance  is 
R'  +(Nf  JN'JR",  according  to  Article  98,  where  P  is  the 


\<7 


O 


Fig.  199. 


primary  inductance  equivalent  of  magnetic  leakage.  The  two 
electromotive  force  components  RI'  and  coPf  are  at  right  angles 
to  each  other,  and  therefore  the  termini  of  RI'  and  of  toPI'  trace 
the  circle  C'  C' .  But  a>P  is  constant,  so  that  the  vector  /' 


THE   INDUCTION    MOTOR. 


257 


is  proportional  to  the  vector  wPI1 ',  and  the  angles  fi,  ft  are  equal. 
Therefore  the  terminus  of  I'  traces  the  circle  CC  as  the  second- 
ary resistance  varies.  The  current  here  considered  is  the  primary 
current  over  and  above  the  magnetizing  current  M.  In  case  the 
magnetizing  current  is  considered  the  locus  of  /'  is  as  shown  in 
Fig.  200,  in  which  A  is  the  part  of  the  primary  current  above 
considered,  Mis  the  magnetizing  current,  /'  is  the  total  primary 
current,  and  Er  is  the  primary  impressed  electromotive  force. 
The  part  A  of  the  primary  current  is,  for  a  I  :  I  transformer, 
equal  and  opposite  to  the  secondary  current  /",  as  indicated. 

The  detailed  electromagnetic  action  of  the  induction  motor  run- 
ning at  a  slip  s  with  a  fixed  value  r2  of  secondary  resistance  is  fully 
represented,  when  the  motor  is  stopped,  by  increasing  its  secondary 
resistance  to  rjs.  For,  when  the  motor  is  stopped,  the  electro- 
motive force  in  the  secondary  becomes  I  /s  times  as  great,  so  that 
secondaiy  current,  primary  current,  and,  in  fact,  every  detail  of 
electromagnetic  behavior  remains  unchanged,  except  that  there  is 
a  negligible  increase  of  eddy  current  and  hysteresis  loss  in  the 


0 

Fig.  200. 

rotor.  In  consequence  of  this  fact,  the  behavior  of  the  induc- 
tion motor  may  be  studied  by  means  of  the  diagram,  Fig.  200. 
Let  us  consider  the  application  of  this  diagram  to  an  induc- 
tion motor,  of  which  the  primary  and  secondary  turns  are  equal. 
We  will  use  the  same  notation  as  in  Article  166.  Draw  a  line  rep- 
resenting E' .  Lay  off  the  line  representing  M  [=  E ' 


258  ELEMENTS   OF   ALTERNATING   CURRENTS. 

The  terminus  of  Mis  one  point  on  our  circle.  From  the  termi- 
nus of  M  draw  the  line  PD  at  right  angles  to  Er .  At  standstill 
the  primary  equivalent  resistance  of  the  motor  is  rl  -\-  rv  and  the 
corresponding  value  of  primary  current  is 

"  ^  +  r2  +  jx 

Knowing  rv  r2  and  x,  this  current,  together  with  M,  may  be  laid 
off,  giving  another  point  on  the  circle.  The  circle  is  thus  deter- 
mined, since  its  center  must  lie  on  the  line  PD. 

To  calculate  a  set  of  corresponding  values  of  /',  I" ,  T,  s, 
power  intake,  power  output  and  power  factor,  proceed  as  follows. 
Choose  a  point  on  the  circle  and  scale  off  the  values  of  I'  and 
/".  Also  scale  off  the  vertical  component  of  /'.  This  compo- 
nent of  /'  multiplied  by  R1  gives  total  power  delivered  to  the 
motor.  From  this  subtract  core  losses  (represented  by  the  cur- 
rent M)  and  resistance  loss  I'\r  This  gives  power  delivered 
to  the  rotor.  This  power  delivered  to  the  rotor  is  all  lost  in  the 
resistance  rjs.  It  is  therefore  equal  to  I"  rjs,  from  which  smay  be 
calculated.  Of  the  total  power  delivered  to  the  rotor,  viz.,  I"  rjs, 
the  fractional  part  s  or  I"  r2  is  lost  in  rotor  resistance  and  the 
fractional  part  I  —  s  or  I"  r2  ( I  —  s)/s  is  mechanical  output. 

Remark. — In  this,  as  in  all  previous  discussions  of  the  trans- 
former and  the  induction  motor,  the  mutually  disturbing  influ- 
ences of  coil  resistances,  magnetizing  current  and  magnetic  leak- 
age are  ignored,  that  is,  the  effect  of  each  of  these  things  is  con- 
sidered to  be  that  which  would  exist  if  the  transformer  were 
ideal  except  for  this  one  thing.  The  mutually  disturbing  influ- 
ence of  magnetizing  current  and  magnetic  leakage  is  partly 
allowed  for  by  making  the  line  /",  Fig.  200,  represent  the  sec- 
ondary current  to  a  slightly  greater  scale  (more  amperes  per 
inch)  than  that  to  which  the  line  /'  represents  the  primary  cur- 
rent. In  short,  if  /'  represents  primary  current  one  ampere  per 
inch,  then  I"  should  represent  secondary  current  v  amperes  per 
inch,  where  v  is  the  number  of  amperes  in  a  rotor  circuit  required 


THE    INDUCTION    MOTOR.  259 

to  balance  in  the  stator  the  magnetizing  action  of  one  ampere  in 
a  stator  circuit. 

PROBLEMS. 

1 1 6.  The  winding  of  an  ordinary  ring- wound  direct-current 
armature  has  24  sections.     These  sections  are  disconnected  from 
the  commutator  and  numbered  in  order  from  I  to  24.     Specify 
the  manner  in  which  these  24  coils  are  to  be  connected  to  two- 
phase  mains  to  produce  in  the  ring  a  rotating  state  of  magnetism 
with  four  poles,  with  six  poles,  and  with  twelve  poles.     Specify 
the  connections  of  the  24  coils  to  three-phase  mains  to  produce 
in  the  ring  a  rotating  state  of  magnetism  with  two  poles,  with 
four  poles,  and  with  eight  poles.     State  the  speed  of  the  magnet- 
ism in  each  case  in  revolutions  per  second,  the  frequency  of  the 
polyphase  currents  being  60  cycles  per  second. 

Sample  answer  for  ^.-pole 3 -phase  connections. — Phase  A  is  con- 
nected as  follows  :  Positive  main  to  -}-  (section  i)  —  to  -f  (sec- 
tion 2)  —  to  —  (section  7)  +  to  —  (section  8)  -f  to  -j-  (section  13) 

—  to  +  (section  14)  —  to  —  (section  19)  -j-  to  —  (section  20)  -f 
to  negative  main. 

Phase  B  is  connected  as  follows  :  Positive  main  to  -f  (section 
3)  —  to  +  (section  4)  —  to  —  (section  9)  -f  to  —  (section  10)  -f 
to  +  (section  15)  —  to  -f  (section  1 6)  —  to  —  (section  21)  -|-  to 

—  (section  22)  -f  to  negative  main. 

Phase  C  is  connected  as  follows :  Positive  main  to  -f  (section 
5)  —  to  -f  (section  6)  —  to  —  (section  1 1)  -f  to  —  (section  12)  -f- 
to  -+-  (section  17)  —  to  +  (section  1 8)  —  to  —  (section  23)  -f  to 

—  (section  24)  -f  to  negative  main. 

Speed  :  Two-phase,  four  poles  30  revolutions  per  second, 
six  poles  20  revolutions  per  second, 
twelve  poles  10  revolutions  per  second. 
Three-phase,  two  poles  60  revolutions  per  second, 
four  poles  30  revolutions  per  second, 
eight  poles  1 5  revolutions  per  second. 

117.  The  armature  of  an  ordinary  direct  current  dynamo  is 


260  ELEMENTS    OF   ALTERNATING   CURRENTS. 

short  circuited  between  the  brushes  when  the  resistance  of  the 
armature  circuit  is  0.036  ohm.  The  number  of  armature  con- 
ductors is  260,  and  the  flux  through  the  armature  is  1,500,000 
lines,  which  is  assumed  to  be  invariable.  The  field  magnet  is 
set  rotating  about  the  shaft  as  an  axis  at  a  speed  of  25  revolu- 
tions per  second,  carrying  the  direct-current  brushes  with  it. 
Calculate  the  torque  dragging  upon  the  armature  when  its  speed 
is  24  revolutions  per  second  and  when  its  speed  is  23  revolutions 
per  second.  What  portion  of  the  power  expended  in  driving 
the  field  is  available  at  the  armature  belt,  and  what  portion  is  lost 
in  heating  the  armature,  friction  losses  being  ignored  ? 

Suggestion. — This  problem  may  be  solved  by  use  of  the  ordi- 
nary equations  of  the  direct-current  dynamo. 

Ans.  (a)  Torque  =  4.37  kg. -meters,  96  per  cent,  of  the  power 
is  available  at  the  armature  belt  and  4  per  cent,  is  lost  in  heating- 
the  armature. 

(b)  Torque  =8. 74  kg.-meters,  92  per  cent,  of  the  power  is 
available  at  the  armature  belt  and  8  per  cent,  is  lost  in  heating 
the  armature. 

1 1 8.  An  ideal  three-phase  induction  motor  takes   5  amperes 
of  current  into  each  phase  of  its  A-connected  primary  member 
at  200  volts  and  60  cycles  per  second.     The  rotor,  which  has  a 
three-phase  winding,  A-connected  to  collecting  rings,  supplies  25 
amperes  of  current  to  each  of  three  similar  circuits,  each  having 
a  power  factor  equal  to  0.75.     The  rotor  is  running  at  %  syn- 
chronous speed.     What  is  the  ratio  of  the  stator  to  the  rotor 
turns  ?     What  is  the  rotor  terminal  voltage  ?     What  is  the  total 
intake  of  power  ?     What  is  the  total  electrical  output  of  power  ? 
What  is  the  mechanical  output  of  power?     Ans.   (a)  Ratio  of 
turns  is  5  :  I.     (^)  Rotor  terminal  voltage  is  §-  X  |  X  200  volts. 
(c~)  Total  intake  of  power  is  2,250  watts.     (d)  Total  electrical 
output  is  750  watts,     (e)  Total  mechanical  output  is  1,500  watts. 

1 19.  An  ideal  induction  motor  has  a  three-phase  stator  wind- 
ing, 56  conductors  to  each  phase,  and  a  two-phase  rotor  winding, 


THE   INDUCTION    MOTOR.  26 1 

132  conductors  to  each  phase.  The  machine,  running  at  half 
synchronous  speed,  takes  25  amperes  into  each  of  its  stator 
circuits  at  220  volts.  Required  the  electromotive  force  induced 
in  each  rotor  circuit  and  the  current  flowing  in  each  rotor  circuit. 
Ans.  244.6  volts,  16.85  amperes. 

Suggestion. — In  the  case  of  equi-phase  stator  and  rotor  wind- 
ings the  electromotive  forces  induced  in  each  circuit  of  stator  and 
rotor  respectively  are  proportional  to  the  number  of  stator  and 
rotor  conductors.  When  stator  and  rotor  are  wound  for  different 
numbers  of  phases  this  simple  relation  between  primary  and 
secondary  electromotive  forces  does  not  hold.  Thus  a  given 
intensity  of  rotating  stator  magnetism  will  induce  in  each  circuit 
of  a  three-phase  distributed  stator  winding  only  -j9^\  as  much 
electromotive  force  as  would  be  induced  in  a  concentrated  wind- 
ing having  the  same  number  of  conductors  ;  and  in  each  circuit 
of  a  two-phase  distributed  rotor  winding  only  -f9/oV  as  mucn 
electromotive  force  as  would  be  induced  in  a  concentrated  wind- 
ing having  the  same  number  of  conductors.  (See  Article  84.) 
Therefore  the  electromotive  force  acting  on  each  circuit  of  a 
three-phase  distributed  stator  winding  is  to  the  electromotive 
force  induced  in  a  two-phase  distributed  rotor  winding  as  955  is 
to  901,  the  number  of  conductors  per  circuit  being  the  same  in 
stator  and  rotor. 

Furthermore,  the  effective  magnetizing  action  of  polyphase 
currents  in  a  polyphase  stator  or  rotor  winding  depends  upon  the 
number  of  phases  as  well  as  upon  the  number  of  conductors  and 
strength  of  current.  On  account  of  this  difference  of  effective 
magnetizing  action,  the  stator  and  rotor  currents  must  be  different 
in  order  that  their  magnetizing  actions  may  balance.  This  re- 
lationship between  stator  and  rotor  currents  is,  however,  most 
easily  arrived  at  by  considering  that  the  intake  and  output  are 
equal,  so  that  the  current  in  each  circuit  of  a  two-phase  dis- 
tributed rotor  winding  is  j  x  •§ -J  f  as  great  as  the  current  in 
each  circuit  of  a  three-phase  stator  winding,  the  number  of  con- 
ductors per  circuit  being  the  same  in  stator  and  rotor. 


CHAPTER    XV. 
TRANSMISSION  LINES. 

170.  Introductory. — Power  may  be  transmitted  by  the  pump- 
ing of  water.  If  great  pressure  is  used  a  given  amount  of  power 
may  be  transmitted  by  a  small  flow  of  water  through  a  small 
pipe.  In  every  case,  however,  there  is  a  loss  of  power  on  ac- 
count of  friction  in  the  pipe.  The  smaller  the  pipe  the  greater 
this  loss  and  the  less  the  first  cost ;  the  best  size  of  pipe  is  that 
for  which  neither  the  first  cost  nor  the  continuous  loss  of  power 
by  friction  is  excessive. 

Similarly,  a  given  amount  of  power  may  be  transmitted  by  a 
small  electric  current  through  a  small  wire  by  using  a  large  elec- 
trical pressure  or  electromotive  force.  In  every  case,  however, 
there  is  a  loss  of  power  on  account  of  the  resistance  of  the  wire. 
The  smaller  the  wire  the  greater  this  loss  and  the  less  the  first 
cost  of  the  line  ;  the  best  size  of  wire  is  that  for  which  neither  the 
first  cost  nor  the  continuous  loss  of  power  by  resistance  is  exces- 
sive. 

It  is  only  by  using  very  large  electromotive  forces  that  long 
distance  transmission  lines  may  be  made  at  a  reasonable  cost,  the 
loss  due  to  resistance  being  at  the  same  time  reasonably  small. 
The  highest  electromotive  force  that  can  be  satisfactorily  used 
upon  a  pole  line  exposed  to  the  air  is  about  60,000  volts,  inas- 
much as  the  leakage  from  wire  to  wire  (outgoing  and  returning 
wires)  in  the  form  of  brush  or  spark  discharge  becomes  excessive 
at  higher  electromotive  forces,  unless  the  wires  are  very  large  and 
very  far  apart.  For  transmission  within  a  radius  of  two  or  three 
miles  1,000  and  2,000  volts  are  usually  employed. 

262 


TRANSMISSION    LINES.  263 

171.  Power  loss  and  electromotive  force  loss  in  line. — If,  say,  10 
per  cent,  of  the  power  output  of  a  direct-current  dynamo  is  lost 
in  the  line,  then   10  per  cent,  of  the  electromotive  force  of  the 
dynamo  is  also  lost  in  the  line  and  90  per  cent,  only  is  effective 
at  the  receiving  circuit.     With  alternating  current,  however,  the 
receiving  circuit  may  receive,  say,  90  per  cent,  of  the  power  out- 
put of  the  dynamo,  while  the  effective  electromotive  force  at  the 
receiving  circuit  may  be  more  or  less  than  90  per  cent,  of  the 
electromotive  force  of  the  dynamo.     The  difference  (numerical) 
between  dynamo  electromotive  force  and  the  electromotive  force 
at  the  receiver  circuit  is  called  the  line  drop  and  this  line  drop  is 
of  more  practical  importance  than  the  power  lost  in  the  line,  in- 
asmuch as  nearly  all   receiving  apparatus  needs  to  be  supplied 
with  current  at  approximately  constant  electromotive  force.     This 
is  usually  provided  for  by  over-compounding  the   dynamo  so  as 
to  keep  the  receiver  electromotive  force  constant.     Thus,  if  the 
line  is  designed  to  give  10  per  cent,  drop,  the  dynamo  would  be 
10  per  cent,  over-compounded. 

172.  Line  resistance. — The  resistance  of  a  wire  for  alternating 
currents  may  in  all  practical  cases  be  taken  to  be  the  same  as  the 
resistance  of  the  same  wire  for  direct  current.     The  fact  is,  how- 
ever, that  the  alternating  current  near  the  axis  of  a  wire  lags  in 
phase  behind  the  current  near  the  surface  of  the  wire,  and  the 
resistance  of  the  wire  is  therefore  larger  for  an  alternating  cur- 
rent than  for  a  direct  current* 

173.  Line  reactance. — The  reactance  of  a   transmission    line 
(outgoing  and  returning  wires  side  by  side)  is  greater  the  smaller 
the  wires  and  the  further  they  are  apart,  and  is  proportional  to 
the  length  of  the  line  and  to  the  frequency.     The  accompanying 
table  gives  the  resistance  and  reactance  per  half  mile  of  trans- 
mission line. 

*See  Merritt,  Physical  Review,  Vol.  5,  p.  47. 


264 


ELEMENTS   OF   ALTERNATING   CURRENTS. 


RESISTANCE  AND  REACTANCE  OF  ONE  MILE  OF  WIRE 
OF  TRANSMISSION  LINE)  (EMMET). 


MILE 


Reactance  in  Ohms. 

Size  of 
wire  B   & 

Resis- 

At  60  cycles  per  sec. 

At  125  cycles  per  sec. 

S.  gauge. 

ohms. 

Wires 

Wires 

Wires 

Wires 

Wires 

Wires 

12  inches 

18  inches 

24  inches 

12  inches 

18  inches 

24  inches 

apart. 

apart. 

apart. 

apart. 

apart 

apart. 

OCOO 

•259 

.508 

•557 

•591 

.06 

•17 

•23 

000 

.324 

•523 

•573 

.607 

.09 

.20 

.26 

00 

.412 

•534 

.588 

.6l8 

.12 

23 

.29 

0 

•519 

•550 

•  603 

•633 

.15 

.26 

•32 

I 

2 

.655 
.826 

.565 
.580 

.614 
.629 

.648 
.663 

.18 
.21 

.28 
•31 

iff 

3 

I.O4I 

•591 

.644 

.674 

.24 

•34 

.41 

4 

I-3I3 

.606 

.656 

.690 

.26 

•37 

•44 

5 

1.656 

.620 

.670 

.704 

•3° 

.40 

•47 

6 

2.088 

•633 

.685 

.720 

•32 

•43 

•49 

7 

2-633 

.647 

.700 

.730 

•35 

.46 

•52 

8 

3-320 

.662 

.712 

•  742 

•38 

•  48 

•  55 

9 

4.186 

.677 

.727 

•76l 

.41 

•5i 

•58 

10 

5.280 

.688 

.742 

.776 

•44 

•54 

.62 

174.  Line  capacity. — The  two  wires  of  a  transmission  line  con- 
stitute a  condenser  which  is  charged  and  discharged  as  the  elec- 
tromotive force  between  the  wires  alternates.  The  current  which 
is  delivered  to  the  line  by  the  generator  when  the  receiving  cir- 
cuit is  disconnected  is  called  the  charging  current  of  the  line.  It 
is  nearly  90°  ahead  of  the  generator  electromotive  force  in  phase. 

The  capacity  in  farads  per  mile  of  a  two-wire  line  is  approxi- 
mately 


4.52  x  io 


C. 


in  which  r  is  the  radius  of  each  wire  and  d  is  the  distance  of  the 
wires  apart  from  center  to  center. 

The  combined  effect  of  line  resistance,  line  reactance  and  line 
capacity,  is  quite  complicated  and  its  full  discussion  is  beyond 
the  scope  of  this  text. 

An  instructive  step-by-step  graphical  solution  of  the  general 
problem  involving  also  leakage  from  line  to  line  is  given  by 
Steinmetz.  (See  "  Alternating  Current  Phenomena,"  third  edi- 


TRANSMISSION   LINES.  265 

tion,  pages  47  to  51.)  The  complete  algebraic  solution  of  the 
problem  is  given  by  Steinmetz.  (See  "Alternating  Current  Phe- 
nomena," third  edition,  pages  163  to  192.) 

When  it  is  necessary  for  practical'  purposes  to  consider  line 
capacity  as  well  as  line  resistance  and  line  reactance,  the  capacity 
effect  of  the  line  may  be  sufficiently  well  represented  by  conden- 
sers located  at  one  or  more  points  along  the  line.  The  problem 
is  thus  simplified,  and  can  be  solved  with  the  help  of  the  formulae 
for  series  and  parallel  connections  as  outlined  in  Chapter  VII.  of 
this  text.  (See  Steinrrfetz,  "  Alternating  Current  Phenomena," 
third  edition,  pages  158  to  163.  Also  see  F.  A.  C.  Perrine  and 
F.  G.  Baum,  Transactions  American  Institute  of  Electrical  Engi- 
neers, Vol.  XVII.,  June  and  July,  1900.) 

The  remainder  of  this  chapter  is  devoted  to  the  comparatively 
simple  problem  of  the  influence  of  line  resistance  and  line  react- 
ance upon  the  electromotive  force  drop  in  alternating-current 
transmission  lines.  The  method  here  outlined  is  sufficiently 
accurate  for  all  practical  calculations  on  short  lines. 

175.  Interference  of  separate  transmission  lines. — When  more 
than  one  transmission  line  (more  than  two  wires)  is  strung  on  the 
same  poles  the  alternating  current  in  each  line  induces  elec- 
tromotive forces  in  the  other  lines  and  affects  the  line  drop. 
This  interference  of  one  line  upon  another  is  obviated  by  cross- 
ing the  lines  at  every  second  or  third  pole,  as  shown  in  Figs. 
201,  202  and  203.  Fig.  201  shows  the  arrangement  of  a  single- 
phase  alternating-current  line  to  avoid  inductive  effects  upon  any 


X          X 


Fig.  201. 


X  X 


X 


Fig.  202. 


266  ELEMENTS   OF   ALTERNATING   CURRENTS. 

other  lines  that  may  be  in  the  neighborhood ;  Fig.  202  shows  the 
arrangement  of  four  wires  for  transmitting  two-phase  currents, 


Fig.  203. 

and  Fig.  203  shows  the  arrangement  of  three  wires  for  transmit- 
ting three-phase  currents. 

Remark. — Transmission  lines  also  affect  neighboring  lines  by 
charging  and  discharging  them  electrostatically  with  the  pulsa- 
tions of  electromotive  force  ;  and  by  leakage  currents  due  to  in- 
complete insulation. 

176.  Calculations  of  a  transmission  line  to  give  a  specified  line 
drop  (single -phase). — A  transmission  line  is  usually  designed  to 
deliver  a  prescribed  amount  of  power  P  at  prescribed  electromo- 
tive force  E  to  a  receiver  circuit  of  which  the  power  factor,  cos  0 
(see  Article  52),  is  given.  The  line  drop,  frequency,  length  of 
line  and  distance  apart  of  wires  are  also  given. 

The  generator  electromotive  force  EQ  is  equal  to  the  sum 
(numerical  sum)  of  E  and  line  drop. 

The  full  load  current  I  is  found  from  El  cos  6  =  P. 

The  component  of  E  parallel  to  /  is  E  cos  6,  and  the  compo- 
nent of  E  perpendicular  to  I  is  E  sin  6. 

By  treating  the  problem  at  first  as  a  direct-current  problem  the 
approximate  resistance  rf  of  the  line  is  found,  namely,  r'  1=  line 
drop.  From  this  approximate  resistance  and  length  of  line,  the 
approximate  size  of  wire  and  line  reactance  x  are  found  from  the 
table  ;  and  since  the  line  reactance  varies  but  little  with  size  of 
wire  the  value  of  x  need  not  be  further  appropriated. 

The  component  of  EQ  parallel  to  7  is  E  cos  9  +  rl  where  r  is 
the  true  resistance  of  the  line,  and  the  component  of  EQ  perpen- 
dicular to  /  is  E  sin  0  +  xl.  Therefore 

E*  =  (E  cos  6  +  r/)2  +  (E  sin  6  +  xl)2 


TRANSMISSION    LINES.  267 

or 


*  -  (E  sin  6  +  */)2  -  E  cos  0 

From  this  equation  the  true  line  resistance  r  may  be  found 
and  thence  the  correct  size  of  wire. 

Example  : 

E  =  20,000  volts 

/>=  1,000  kilowatts 

cos  6  =  .85  =  power  factor  of  receiving  circuit 

EQ  =  23,000  volts,  or  line  drop=  3,000  volts 

frequency  =  60  cycles  per  second 

distance  =  30  miles 

distance  apart  of  wires  =  1 8  inches 
From  these  data  we  find  : 

/=  58.8  amperes 

V  =  5 1  ohms 

Therefore,  from  the  table  we  find  that,  approximately,  a  No. 
2  B.  &  S.  wire  is  required  so  that  x  =  37.7  ohms. 
Further 

E  cos  6  —  1 7,000  volts 

E  sin  6  +  xl  =.  12,700  volts 
and  from  equation  ( 1 26)  we  find 

r  =  37.3  ohms 

from  which  the  correct  size  of  wire  is  found  to  be  approximately 
a  No.  i  B.  &  S. 

177.  Calculation  of  double  line  for  two-phase  transmission  (four 
wires). — In  this  case  each  line  is  calculated  to  deliver  half  the 
prescribed  power.  Thus,  if  it  is  desired  to  deliver  7,000  kilo- 
watts at  20,000  volts  two-phase,  at  a  frequency  of  60,  line  drop 
of  3,000  volts,  etc.,  then  each  line  is  calculated  as  a  single-phase 
line  to  deliver  500  kilowatts  at  3,000  volts  line  drop,  the  lines 
being,  of  course,  arranged  as  shown  in  Fig.  202. 


268  ELEMENTS   OF   ALTERNATING   CURRENTS. 

178.  Calculation  of  a  three-wire  transmission  line  for  three- 
phase  currents, — The  calculation  will  be  carried  out  for  the  case 

of  Y-connected  generator 
Generator  j  Receiver  ,  Tr 

and  Y-connected  receiver 

as  shown  in  Fig.  204,  for 
the  reason  that  the  rela- 
tion between  EQ,  E,  and 
line  current  is  then  the 
Fig- 204'  simplest. 

Let  cos  6  be  the  power  factor  of  each  receiving  circuit,  P  tne 
total  power  to  be  delivered,  E  the  electromotive  force  between 
the  terminals  of  each  receiving  circuit  and  EQ  the  electromotive 
force  of  each  armature  winding  on  the  generator  ;  *  all  prescribed. 

Then  P—  ^EI cos  6 

from  which  the  full  load  line  current  /  may  be  calculated. 

The  difference  E0  —  E  is  due  to  electromotive  force  drop  in  one 
main.  Therefore,  looking  upon  the  problem  as  one  in  direct 
currents,  we  have  E0  —  E  =  r'/ where  r'  is  the  approximate  re- 
sistance of  one  main.  From  this  the  approximate  size  of  wire 
may  be  found  from  the  table. 

Consider  one  of  the  mains  ;  say,  main  No.  2  ;  the  other  two 
mains  together  constitute  the  return  circuit  for  this  main,  and  the 
average  distance  from  main  2  to  mains  i  and  3  is  \\l  when  the 
mains  are  crossed  as  shown  in  Fig.  203.  Find  the  reactance  x 
of  a  pair  of  mains  each  of  the  size  approximated  above  and  dis- 
tant I  \l  from  each  other. 

The  component  of  E  parallel  to  /  is  E  cos  6  and  the  com. 
ponent  of  E  perpendicular  to  f  is  E  sin  6. 

The  resistance  drop  in  one  main  is  rl  and  the  reactance  drop 
in  one  main  is  \xl,  the  former  being  parallel  to  /  and  the  latter 
perpendicular  to  /. 

Then  the  components  of  E0  are  E  cos  0  +  rl  and  E  sin  6  -f  \xl 
so  that 

E*  =  (E  cos  6  -f  r/)2  +  (E  sin  6  +  fr/)» 

*  The  electromotive  force  between  mains  at  receiving  station  is  V$E  and  the  elec- 
tromotive force  between  mains  at  generating  station  is  V~3  EQ. 


TRANSMISSION   LINES.  269 

or 

*  si"  *  +  -  E  cos  9 


_ 

which  gives  r,  the  true  resistance  of  one  main,  from  which  the 
correct  size  of  wire  is  easily  found. 

The  calculation  of  a  transmission  line  when  the  electromotive 
forces  between  mains  is  specified  instead  of  the  electromotive 
forces  in  Y-connected  circuits  is  sufficiently  explained  in  the  fol- 
lowing example. 

Example.  —  Electromotive  force  between  mains  at  receiving 
station  to  be  20,000  volts.  Therefore,  electromotive  force  be- 
tween terminals  of  Y-connected  receiving  circuits  would  be 
20,000  -h  V  3.  Therefore 

£=  1  1,550  volts 

Electromotive  force  between  mains  at  generating  station  to  be 
23,000  volts.  Therefore,  EQ  =  22,000  -=-  v/3,  or 

£0=  13,280  volts 
Further  specifications  : 
P=  1,000  kilowatts, 
cos  #  =  .85, 

frequency  =  60  cycles  per  second, 
distance  =  30  miles, 

distance  apart  of  adjacent  wires  =15^  inches  (/). 
From  these  data  we  find 

7=  34.0  amperes 
r'  =  50.9  ohms 

Therefore,  approximately,  a  No.  5  wire  is  needed.  The  react- 
ance, x,  of  a  30-mile  double  line  of  No.  5  wire  at  2  1  inches 
(=  1  1^/)  apart  is,  from  the  table, 

^=41.2  ohms 

V 

Equation  (127)  then  gives 

r  =  46.  5  ohms 

So  that  a  wire  between  No.  4  and  No.  5  would  give  the 
prescribed  line  drop. 


ALTERNATING  CURRENT 
MACHINERY. 


CHAPTER   XVI. 
ALTERNATORS. 

179.  When  alternators  are  to  be  used  solely  for  the  operation 
of  incandescent  lights,  arc  lamps,  or  in  general  any  device  in 
which  the  heating  effect  of  the  current  is  made  use  of,  single- 
phase  machines  are  suitable.      They  are  simpler  than  two-  or 
three-phase  machines,  and  perform  the  work  equally  well.      In 
fact,  in  some  particulars  they  are  to  be  preferred,  as  they  do 
not  give  rise  to  the  unbalanced  voltages  often  met  with  in  poly- 
phase working.      In  most  modern  plants,  however,   motors   as 
well  as  lights  are  operated  and  if  the  motor  load  is  at  all  large, 
it  is  best  to  install  polyphase  alternators. 

A  three-phase  or  two-phase  alternator  can  be  used  to  operate 
single-phase  circuits  by  distributing  the  circuits  so  that  the  load 
on  the  different  phases  will  be  approximately  balanced.  A  three- 
phase  or  two-phase  alternator  costs  but  little  more  than  a  single- 
phase  machine  of  similar  output,  so  that  it  is  quite  common 
practice  to  install  polyphase  machines  even  if  they  are  operated 
for  the  time  being  on  single-phase  circuits,  because  should  it  be 
necessary  to  operate  motors  later,  the  polyphase  currents  will 
then  be  available.  On  account  of  these  features,  the  number  of 
single-phase  machines  now  installed  is  very  small  as  compared 
with  polyphase. 

180.  Construction  of  alternators. — So  far  as  general  construc- 
tion and  appearance  are  concerned,  single-phase  and  polyphase 
alternators  are  practically  identical.      About  the  only  difference 
lies  in  the  arrangement  and  connections  of  the  armature  winding 
and  the  number  of  collector  rings,  with  which  the  machines  are 
provided.     Three  general  classes  of  alternators  are  in  common 

273 


274  ALTERNATING   CURRENT   MACHINERY. 

use,   as   follows:    these  have  already  been   briefly   described   in 
Chapter  II. 

(a)  Machines  with  a  stationary  field  and  revolving  armature. 
In  these  machines  the  current  is  led  from  the  armature  by  means 
of  collector  rings,  and  the  conductors  in  which  the  electromotive 
force  is  induced  are  mounted  rigidly  on  the  armature  core  and 
revolve  with  it. 

(b)  Machines  with  a  stationary  armature  and  revolving  field. 
In  these  alternators  the  armature  windings  are  arranged  in  slots 
around  the  inner  periphery  of  a  stationary  armature  structure. 
The  revolving  field  is  usually  provided  with  radially  projecting 
poles  and  revolves  within  the  stationary  armature.     The  exciting 
current  is  led  into  the  field  windings  by  means  of  two  collector 
rings  on  the  shaft  and  the  lines  leading  from  the  machine  connect 
directly  to  the  armature  winding.     This  type  of  machine  has  of 
late  years  come  into  very  extensive  use,  especially  for  alternators 
of  large  size  and  high  voltage  or  large  current  output.      The 
revolving  construction  allows  the  armature  to  be  insulated   for 
very  high  voltages,  it  being  possible  to  build  such  machines  to 
generate  pressures  of  twelve  thousand  volts  or  even  higher. 

(c)  Machines  in  which  a  revolving  mass  of  iron  with  polar  pro- 
jections causes  the  magnetic  flux  passing  through  a  set  of  sta- 
tionary coils  to  vary,  thus  setting  up  an  alternating  electromotive 
force  of  them.     In  these  machines,  known  as  inductor  alternators, 
there  is  no  moving  wire.      The  armature  coils  are  stationary, 
being  arranged  in  very  much  the  same  way  as  those  for  a  re- 
volving field  machine.     The  magnetic  flux  is  set  up  by  a  station- 
ary field  coil,  and  while  the  flux  passing  through  the  armature 
coil  varies  from  zero  to  a  maximum,  it  does  not  reverse  as  in  the 
revolving  field  or  revolving  armature  machines.     This  descrip- 
tion applies  to  the  Stanley  alternator  which  is  the  most  prominent 
example  of  this  type  used  in  America. 

181,    Revolving  armature  alternators. — Fig.  205  shows  a  West- 
inghouse  3OO-kilowatt,  I33~cycle,  belt-driven,  single-phase  alter- 


ALTERNATORS. 


275 


nator  of  the  revolving  armature  type.  A  machine  of  this  type 
would  be  suitable  for  lighting  or  heating  work  only.  Frequen- 
cies as  high  as  133  cycles  per  second  are  now  seldom  used,  but 


Fig.  205  will  serve  as  an  illustration  of  a  belt-driven  type  of 
alternator.  The  lower  half  of  the  field  yoke  is  in  one  casting 
with  the  bearings.  The  poles,  of  which  there  are  thirty-two 


2/6  ALTERNATING   CURRENT   MACHINERY. 


Fig.  206a. 


Fig.  206b. 


ALTERNATORS.  277 

project  radially  inwards,  and  are  laminated,  each  pole  piece  being 
made  up  of  a  number  of  sheet-iron  stampings  firmly  held  together 
between  end  plates.  These  pole  pieces  are  cast-welded  into  the 
yoke.  Fig.  206  (a)  shows  the  armature  core  and  a  number  of 
the  coils  for  a  single-phase  machine  of  the  belt-driven  type,  and 
Fig.  206  (b)  shows  a  revolving  armature  of  large  diameter  for 
a  slow-speed  direct-connected  machine.  It  should  be  noted  that 
the  winding,  Fig.  206  (a),  is  of  the  partially  distributed  type, 
there  being  three  coils  pei;  pole  arranged  one  inside  of  the  other 
as  indicated  by  the  group  of  coils  at  the  left.  The  winding  on 
the  armature,  Fig.  206  (b),  is  also  partially  distributed,  there 
being  ten  slots  per  pole.  On  the  end  of  the  shaft,  Fig.  205,  out- 
side of  the  right-hand  bearing,  is  shown  the  rectifier  for  supplying 
current  to  the  series  field  coils.  The  rectifier  connects  to  the 
secondary  of  a  small  series  transformer  carried  on  the  armature 
spider,  the  primary  of  the  transformer  being  in  series  with  the 
armature  winding.  (See  Article  94.)  Alternators  with  re- 
volving armatures  are  made  in  a  large  range  of  sizes,  and  may 
be  either  of  the  belt-driven  or  direct-connected  type.  However, 
the  revolving  armature  alternator  is  becoming  obsolete  since  for 
most  purposes  it  is  not  as  desirable  as  the  revolving  field  type. 

182.  Revolving  field  alternators. — Fig.  207  (a)  shows  one  ar- 
rangement of  the  field  and  armature  of  machines  of  this  class. 
The  armature  coils  a  are  held  in  the  slots  b  arranged  around  the 
inner  periphery  of  the  core  C,  which -is  built  up  of  stampings 
clamped  between  plates  D  and  supported  by  the  frame  M. 
The  revolving  field  consists  of  radial  laminated  pole  pieces  E, 
carrying  exciting  coils  F.  These  pole  pieces  are  bolted  to  the 
steel  yoke  or  rim  G,  which  is  connected  to  the  shaft  by  means  of 
the  field  spider  arms  H.  The  exciting  coils  F  are  wound  with 
a  copper  strip,  which  is  coiled  on  edge  as  shown.  *  This  makes 
a  very  solid  and  substantial  coil  and  one  from  which  the  heat  is 
readily  radiated.  Fig.  207  (a)  shows  the  general  construction 
of  the  revolving  field  alternators  built  by  the  General  Electric 


278 


ALTERNATING   CURRENT    MACHINERY. 


Co.  The  Westinghouse  revolving  field  is  constructed  as  shown 
in  (b)  ;  the  rim  and  pole  pieces  are  here  built  up  of  stampings 
a  which  are  staggered  or  overlapped  so  as  to  form  a  continuous 
structure.  The  exciting  coils  b  are  held  from  flying  out  by  metal 
bridges  c  fitted  into  the  notches  in  the  pole  pieces.  These  bridges 


Fig.  207. 

help  to  suppress  hunting  by  virtue  of  the  eddy  current  generated 
in  them  when  there  is  a  momentary  angular  variation  in  speed. 
Fig.  208  sho'ws  a  General  Electric  alternator  of  the  revolving 
field  type  and  having  the  general  construction  indicated  in  Fig. 
207  (a).  The  two  collector  rings  mounted  on  the  shaft  are  for 
supplying  the  exciting  current.  In  this  type  of  machine  the  sta- 


ALTERNATORS. 


2/9 


tionary  armature  structure  is  usually  arranged  so  that  it  can  be 
slid  to  one  side  on  the  bed  plate,  thus  allowing  access  to  both 
armature  and  field  coils. 

Since  the  armature  windings  are  stationary  on  revolving  field 
machines,  there  are  no  collector  rings  to  insulate  for  high  pres- 
sures. Also,  there  is  plenty  of  available  room  for  thorough  insu- 


Fig.  208. 

lation,  so  that  the  armature  can  readily  be  insulated  for  compara- 
tively high  pressure.  Machines  of  this  type,  of  3,500  k.  w. 
capacity,  three-phase,  are  used  for  operating  the  surface  street- 
car system  in  New  York  City.  These  alternators  are  direct-con- 
nected to  vertical  engines  running  at  75  revolutions  per  minute. 
The  alternators  generate  6,600  volts,  and  supply  current  to  a 


280  ALTERNATING   CURRENT    MACHINERY. 


ALTERNATORS.  281 

number  of  substations,  where  the  pressure  is  stepped  down  by 
means  of  transformers.  It  is  then  supplied  to  rotary  converters 
for  conversion  into  direct  current  at  500  volts  for  operating  the 
cars. 

Fig.  209  shows  sectional  vjews  of  a  Bullock  three-phase  600- 
kilowatt  alternator.  It  has  18  poles  and  runs  at  a  speed  of  400 
revolutions  per  minute.  The  machine  therefore  delivers  current 
having  a  frequency  of  60  cycles  per  second.  The  revolving  field 
is  75^  inches  outer  diameter,  so  that  the  peripheral  speed  is 
nearly  8,000  feet  per  minute.  The  pole  pieces  are  made  up  of 
laminations  and  are  dovetailed  into  the  rim  of  the  field  spider, 
which  in  this  case  is  of  cast  steel  and  is  in  the  form  of  a  continu- 
ous disc  instead  of  a  rim  provided  with  spokes.  The  whole  field 
construction  is  therefore  such  as  to  withstand  the  heavy  cen- 
trifugal strains  to  which  it  is  subjected  on  account  of  the  high 
peripheral  speed.  The  stationary  armature  is  provided  with  two 
slots  per  pole  per  phase,  so  that  the  winding  is  partially  dis- 
tributed. The  exciting  current  is  led  into  the  revolving  field  by 
means  of  the  two  collector  rings  shown  mounted  on  the  shaft. 
The  field  coil  conductor  consists  of  copper  strips  wound  on  edge. 

183.  Inductor  alternators. — The  Stanley  alternator  is  one  of 
the  most  prominent  examples  of  the  inductor  type.  The  general 
arrangement  of  this  machine  is  shown  in  Fig.  210.  The  machine 
is  double,  and  has  practically  two  independent  armatures,  A,  A'. 
The  two  armature  cores  are  connected1  by  the  bars  b,  which 
carry  the  magnetic  flux,  and  the  armature  coils  cc  are  arranged 
around  the  inner  periphery  of  the  laminated  core  structure  in 
practically  the  same  way  as  for  a  revolving  field  machine.  The 
revolving  inductor  dd  carries  a  crown  of  projecting  laminated 
poles  pieces  pp  at  each  end,  but,  unlike  the  revolving  field  ma- 
chine, these  polar  projections  are  not  alternately  of  opposite  po- 
larity. All  those  at  one  end  of  the  inductor,  such  as  NNNf  are 
of  one  polarity,  and  all  those  at  the  other  end  are  of  opposite 


282 


ALTERNATING   CURRENT   MACHINERY. 


polarity.  The  magnetic  flux  takes  the  path  shown  by  the  curved 
dotted  line  hfgh,  and,  as  the  inductor  revolves,  the  flux  threading 
the  armature  coils  alternately  increases  to  a  maximum  and  de- 
creases to  zero,  but  does  not  reverse.  The  flux  is  set  up  by  a 
large  stationary  coil  C,  which  completely  encircles  the  inductor. 
The  armature  coils,  as  indicated  in  Fig.  210,  are  arranged  to  gen- 
erate two  electromotive  forces  differing  in  phase  by  90°,  one  set 
of  coils  being  displaced,  so  that  the  electromotive  force  in  them 
is  zero  at  the  instant  the  electromotive  force  in  the  other  coils 


Fig.  210. 

is  a  maximum.  Inductor  alternators  are  also  made  by  the  War- 
ren and  Westinghouse  companies.  Their  machines  differ  from 
the  Stanley  alternator  in  having  only  one  stationary  armature 
instead  of  being  double,  otherwise  the  principle  of  operation 
is  the  same.  Inductor  alternators  are  not  as  largely  used  as 
those  with  a  revolving  field.  They  are  more  expensive  to  build 
than  machines  of  the  revolving  field  type  if  designed  to  give 
equally  good  voltage  regulation. 


ALTERNATORS.  283 

184.  Connections  for  parallel  running  of  alternators. — When 
alternators  are  arranged  for  parallel  running,  some  means  must 
be  provided  for  telling  when  the  machines  are  in  synchronism. 
If  one  alternator  is  running  and  it  is  desired  to  throw  another  in 
parallel  with  it,  the  second  machine  must  first  be  brought  up  to 
synchronism  and  thrown  in  with  the  other  at  the  instant  when 
the  electromotive  forces  of  the  two  machines  are  in  phase.  A 
number  of  different  instruments  have  been  brought  out  for  indi- 
cating the  condition  of  synchronism,  but  in  many  cases  incan- 
descent lamps  (see  Art.  126),  or  a  synchronizing  voltmeter  are 
used.  It  is  essential  that  the  two  alternators  be  running  at 
the  same  frequency  when  they  are  thrown  in  parallel.  A  slight 
difference  in  phase  does  not  make  so  much  difference,  as  the  ma- 
chines will  pull  each  other  into  step,  the  only  disadvantage  being 
a  considerable  exchange  of  current  between  the  machines. 

Fig.  211  shows  the  essential  connections  required  for  the  oper- 
ation of  two  compound  wound  alternators  in  parallel ;  aa  repre- 
sent the  collector  rings  and  rectifier;  bb  are  the  series  field  coils. 
The  separately  excited  field  coils  and  exciter  connections  are 
omitted  in  order  to  avoid  confusing  the  diagram.  When  com- 
pound wound  alternators  are  run  in  parallel,  it  is  necessary  to 
connect  the  series  coils  in  parallel,  so  that  the  current  can  equal- 
ize between  them.  This  is  accomplished  by  tHe  equalizing  wires 
dd  running  between  the  machines.  Adjustable  compounding  re- 
sistances rrr  are  connected  in  parallel  with  the  series  coils,  so  that 
the  effect  of  the  coils,  and,  hence,  the  degree  of  compounding 
can  be  varied  by  shunting  a  portion  of  the  current  that  would 
otherwise  flow  through  them.  The  collector  rings  are  connected 
through  the  main  switch,  to  the  bus-bars.  In  Fig.  211  three- 
phase  alterations  are  shown  and  the  ammeter  is  indicated  in  but 
one  of  the  line  wires.  If  the  system  is  well  balanced  only  one 
ammeter  is  necessary,  but  if  unbalancing  is  probable  an  ammeter 
is  placed  in  each  of  the  three  wires.  TT'  are  small  potential 
transformers  to  step  down  the  pressure  for  the  synchronizing 
lamps  //'  and  voltmeters  VV .  These  transformers  are  connected 


284 


ALTERNATING   CURRENT    MACHINERY. 


back  of  the  main  switch,  i.  e.,  between  the  main  switch  and  the 
alternator,  because  it  is  necessary  to  know  the  voltage  and  phase 
relation  of  any  machine  before  the  main  switch  is  thrown  in 
to  connect  the  machine  to  the  bus-bars.  When  the  synchro- 
nizing plugs  are  inserted  at  PP',  the  secondaries  of  TT'  are  con- 
nected in  series  with  the  lamps  //'.  If  the  connections  are  such 
that  the  electromotive  forces  of  T  and  T'  oppose  each  other  when 
the  machines  are  in  step,  lamps  IV  will  be  dark  when  synchronism 
is  attained,  but  if  the  electromotive  forces  in  the  synchronizing 
circuit  are  directed  as  indicated  by  the  arrowheads  in  Fig.  211 


Fig.  211. 


the  lamps  will  burn  at  full  candle  power  at  synchronism  because 
the  electromotive  forces  then  aid  each  other.  In  some  cases  the 
connections  are  made  so  that  dark  lamps  indicate  synchronism; 
in  other  cases  light  lamps  are  used.  The  latter  is  probably  pref- 
erable, because  it  is  easier  to  tell  the  point  of  maximum  bright- 
ness than  maximum  darkness.  When  the  pulsations  of  the  lamps 
become  very  slow,  say  one  beat  in  two  or  three  seconds,  it  indi- 
cates that  the  machines  are  running  at  very  nearly  the  same 
frequency,  and  the  main  switch  should  be  closed  in  the  middle 


ALTERNATORS. 


285 


of  one  of  the  beats  when  the  lamps  are  light  or  dark,  as  the  case 
may  be. 

Fig.  212  shows  one  scheme  of  connections  used  by  the  West- 
inghouse  Company  for  running  two  three-phase  machines  in 
parallel.  The  exciter  and  field  connections  are  here  omitted  for 
the  sake  of  simplicity.  The  machines  are  separately  excited,  no 
series  winding  being  used  on  the  field.  Plain  separate  excitation 
is  used  on  most  of  the  larger  alternators  now  installed.  By  care- 


'tmeter 


tfach/ne  fl?l 


Fig.  212. 


ful  designing,  the  voltage  regulation  can  be  made  sufficiently 
close,  so  that  whatever  regulation  is  necessary  can  be  obtained  by 
varying  the  separate  field  excitation.  In  Fig.  212,  therefore,  no 
equalizing  connections  are  necessary.  The  lines  from  the 
machines  lead  first  to  the  main  switch  and  then  through  the  am- 
meters and  connect  to  the  main  bus-bars.  One  ammeter  is  pro- 
vided in  each  line  although  this  is  not  absolutely  necessary  under 
ordinary  circumstances.  One  voltmeter  is  made  to  serve  for  any 
number  of  machines  by  providing  a  set  of  plug  receptacles  A,  A, 


286  ALTERNATING   CURRENT    MACHINERY. 

for  each  machine.  T  and  T  are  potential  transformers,  one  for 
each  alternator,  and  connected  back  of  the  main  switch.  T"  is 
a  potential  transformer  connected  to  the  bus-bars.  By  inserting 
a  plug  in  the  right-hand  receptacle,  as  indicated  by  the  dotted 
lines  at  A,  the  voltage  of  machine  No.  I  is  indicated.  By  insert- 
ing the  plug  in  the  left-hand  receptacle,  the  voltmeter  indicates 
the  bus-bar  voltage.  The  attendant  can,  therefore,  readily,  com- 
pare the  voltage  of  the  machine  which  is  to  be  thrown  in  parallel, 
with  the  voltage  on  the  bus-bars  to  which  it  is  to  be  connected. 
Pilot  lamps  are  shown  at  aa' ' ,  and  II'  are  the  synchronizing 
lamps.  When  the  synchronizing  plugs  are  inserted  at  pp'  cur- 
rent tends  to  flow  around  the  circuit  indicated  by  the  arrow- 
heads. If  the  machines  were  in  phase,  the  electromotive  forces 
of  T  and  T'  would  oppose  each  other  with  the  connections 
shown,  and  the  lamps  //'  would  be  dark  at  synchronism.  By 
reversing  the  connections  of  one  of  the  transformers,  the  trans- 
former electromotive  forces  would  be  added  together  at  synchro- 
nism and  the  two  lamps  would  burn  up  to  full  brightness. 

185.  Fig.  213  shows  connections  for  synchronizing  3-phase  ma- 
chines, due  to  Mr.  J.  E.  Woodbridge.  In  this  case  the  voltmeters 
V^  or  V2  are  used  to  indicate  the  point  of  synchronism,  but  lamps 
are  also  provided.  In  the  ordinary  method  just  described,  the 
voltage  acting  on  the  synchronism-indicating  device  is  the  resul- 
tant of  two  equal  E.M.F.'s  either  in  phase  (bright  lamps)  or 
1 80°  apart  in  phase  (dark  lamps)  and  the  voltage  acting  on  the 
device  varies  from  zero  to  twice  the  normal  voltage.  In  the  con- 
nections shown  -in  Fig.  213  the  voltage  acting  on  the  voltmeter 
is  the  resultant  of  two  equal  E.M.F.'s  differing  in  phase  by  60°. 
This  has  the  advantage  that  the  rate  of  change  of  the  resultant 
E.M.F.  acting  on  the  voltmeter,  with  variation  in  the  phase  rela- 
tion of  the  two  components  is  much  higher  than  when  the  two 
components  are  180°  apart  in  phase.  In  other  words,  a  small 
change  in  phase  relation  of  the  two  alternators  makes  a  consider- 
able change  in  the  voltmeter  reading.  In  Fig.  213,  a,  b,  c,.d  are 
the  primary  coils  of  potential  transformers  and  e,  f,  g,  h  the  sec- 


ALTERNATORS. 


287 


ondaries ;  a  and  c  are  connected  to  corresponding  phases,  as  also 
are  b  and  d.  The  junctions  of  the  secondary  coils  are  grounded 
at  m  and  n;  this  simplifies  the  connections  and  also  protects  the 
secondary  circuit  from  dangerously  high  potential  in  case  of  a 
breakdown  or  accidental  connection  between  the  primary  and 
secondary.  When  plugs  are  inserted  at  k  and  I,  assuming  that 
machine  No.  I  is  running  and  that  No.  2  is  to  be  synchronized, 
secondaries  e  and  h,  which  are  connected  across  different  phases, 
are  in  series  with  the  voltmeter  F2  by  way  of  the  two  ground 
connections.  The  secondaries  are  connected  in  opposition,  so 


To  A/ternatorNal. 


\Pluff  for 
{.Machine  Running 

\  Pfugfpr 
\Fteadirtg  Voltage. 


To  Afternator  No.  2. 
Fig.  213. 


that  the  E.M.F/s  acting  on  voltmeter  F2  differ  in  phase  by  60° 
at  synchronism.  The  connections  of  the  lamps  do  not  require 
comment;  secondaries  e  and  g  are  across  similar  phases  and  the 
lamps  are  dark  at  synchronism.  As  the  machines  come  into  syn- 
chronism the  swing  of  the  voltmeter  F2  becomes  slower  and  the 
switch  is  closed  when  .the  needle  is  at  its  maximum  throw. 

186.  Lincoln  Synchronizer. — With  the  increasing  size  of  alter- 
nators a  need  has  been  felt  for  an  instrument  that  would  be  more 
accurate  than  either  lamps  or  voltmeters  for  showing  the  exact 


288 


ALTERNATING   CURRENT    MACHINERY. 


phase  relation  between  two  machines  that  are  about  to  be  con- 
nected in  parallel.  In  case  a  large  machine  is  connected  to  the 
bus-bars,  even  when  differing  but  slightly  in  phase  or  frequency 
from  the  bus-bar  E.M.F.,  there  will  be  a  large  flow  of  current 
that  may  cause  undesirable  voltage  fluctuations.  It  is  necessary 
to  have  an  instrument  that  will  show  whether  the  incoming 
machine  is  running  too  fast  or  too  slow,  whether  it  is  coming 
into  or  going  out  of  phase,  and  when  it  is  exactly  in  phase.  A 
number  of  synchronism  indicators  or  synchronoscopes  have  been 
devised;  one  that  is  used  in  many  central  stations  in  America 
is  that  due  to  Mr.  Paul  M.  Lincoln  and  shown  in  Fig.  214.  The 
instrument  is  provided  with  four  terminals,  two  of  which  are 


Fig.  214. 

connected  to  the  bus-bars  and  the  other  two  to  whatever  machine 
is  to  be  synchronized.  If  the  pressure  is  over  400  or  500  volts, 
potential  transformers  are  inserted  between  the  synchronizer  and 
the  bus-bars  or  alternator.  If  the  incoming  alternator  is  run- 
ning too  slow,  the  hand  of  the  synchronizer  revolves  to  the  right 
at  a  speed  corresponding  to  the  difference  in  frequency  between 
that  of  the  incoming  machine  and  the  bus-bars.  If  the  alter- 
nator is  running  too  fast,  the  hand  revolves  to  the  right.  The 
attendant  can  thus  tell  at  a  glance  whether  or  not  the  incoming 
machine  should  be  speeded  up  or  slowed  down.  As  the  alter- 


ALTERNATORS. 


289 


nator  comes  more  and  more  nearly  into  synchronism,  the  revolu- 
tions of  the  hand  become  slower  and  slower,  and  when  the 
pointer  is  moving  very  slowly  and  is  near  the  vertical  position  the 
main  switch  is  thrown  in.  Owing  to  the  fact  that  the  instru- 
ment indicates  at  all  times  the  exact  condition  of  the  incoming 
machine  as  regards  its  speed  and  phase  relation,  the  operation 
of  synchronizing  can  be  carried  out 
much  more  quickly  and  with  more 
certainty  than  where  lamps  or  volt- 
meters are  used. 

•  The  operation  of  the  Lincoln 
synchronizer  will  be  understood 
by  referring  to  Fig.  215.  B,  B 
is  a  laminated  field  constructed 
in  the  same  way  as  the  field  of  a 
small  two-pole  motor.  On  this 
field  are  wound  the  coils  S\S  which 
are  connected  to  the  bus-bar  either 
directly  or  through  a  potential 
transformer  depending  on  the  volt- 
age. The  core  A  is  mounted  on  a 
shaft  which  carries  the  pointer  n. 
On  A  are  wound  two  coils  C  and  D 
substantially  at  right  angles  to  each 
other  and  connected  in  series  at  e. 
The  three  terminals  lead  to  collector 
rings  /,  g,  and  h  mounted  on  the 
shaft.  Coil  C  is  connected  in 
series  with  an  inductance  L  and 
coil  D  in  series  with  a  non-inductive  resistance  R.  The  junc- 
tion of  the  coils  is  connected  to  one  terminal  of  the  machine 
transformer  as  shown.  The  current  in  coil  D  is  in  phase  with 
the  machine  E.M.F.,  while  that  in  C  lags  90°  behind  the  ma- 
chine E.M.F.  since  the  inductance  L  is  adjusted  to  give  a  phase 
difference  of  practically  90°.  The  magnetic  flux  set  up  in  the 


Fig.  215. 


290  ALTERNATING   CURRENT    MACHINERY. 

field  BB  will  lag  nearly  90°  behind  the  bus-bar  E.M.F.  because, 
owing  to  the  inductance  of  coils  SS,  the  field  current  will  lag 
nearly  90°  behind  the  E.M.F.  Suppose  that  the  E.M.F.  of  the 
incoming  machine  is  exactly  in  phase  with  the  bus-bar  E.M.F., 
and  that  the  two  are  at  exactly  the  same  frequency.  The  cur- 
rent in  coil  C  will  then  be  in  phase  with  the  magnetic  flux  and 
coil  C  will,  therefore,  assume  the  position  shown  in  the  figure 
and  the  pointer  will  be  stationary  and  vertical.  The  current  in 
D  will  be  90°  out  of  phase  with  the  field  and  there  will  be 
no  torque  action  exerted  on  the  coil.  Suppose  that  the  ma- 
chine E.M.F.  differs  in  phase  by  90°  from  the  bus-bar  E.M.F, 
Then  the  current  in  D  will  be  in  phase  with  the  field  flux  and 
that  in  C  will  differ  in  phase  by  90°  from  the  flux ;  the  armature 
will  then  turn  through  90°  until  coil  D  assumes  the  vertical  posi- 
tion. In  other  words,  a  change  in  phase  relation  is  accompanied 
by  a  corresponding  angular  movement  of  the  pointer.  So  far 
we  have  assumed  that  the  two  frequencies  are  exactly  the  same. 
It  is  evident  that  if  there  is  a  difference  in  frequency,  the  phase 
relations  between  the  currents  in  the  armature  coils  and  the  field 
flux  are  continually  changing  and  consequently  the  hand  revolves 
at  a  speed  corresponding  to  the  difference  in  frequency  because 
the  coils  CD  are  continually  attempting  to  take  up  a  position  cor- 
responding to  the  phase  relation  of  the  two  E.M.F/s.  The  point 
at  which  the  machine  is  connected  in  parallel  is  when  the  pointer 
occupies  a  position  somewhere  within  an  arc  such  as  op  and  when 
the  movement  of  the  hand  is  slow.  Appliances  have  been  devised 
to  close  the  main  switch  automatically  when  the  pointer  is  moving 
slowly  within  the  arc  op,  thus  eliminating  the  operator  entirely. 
So  far,  however,  these  automatic  methods  of  synchronizing  do 
not  appear  to  be  used  very  extensively. 


CHAPTER   XVII. 
TRANSFORMERS. 

187.  Construction. — The  construction  of  a  transformer  depends 
to  a  certain  extent  upon  the  use  to  which  it  is  to  be  put.  Trans- 
formers for  outdoor  use  must  be  protected  by  water-proof  cases, 
and  little  provision  can  be  made  for  ventilation.  On  the  other 


292 


ALTERNATING    CURRENT   MACHINERY. 


hand,  where  transformers  are  used  indoors,  as,  for  example,  in 
substations,  the  question  of  ventilation  and  cooling  becomes  im- 
portant and  the  construction  of  the  transformer  has  to  be  modi- 
fied accordingly. 

Transformers   used   for   outdoor   work   are   of   comparatively 
small  size;  they  are  placed  in  water-proof  iron  cases  which  are 


/0\ 


Fig.  217. 

usually  designed  so  that  they  can  be  filled  with  oil,  though  in 
some  cases,  especially  with  small  transformers,  oil  is  not  used. 
The  oil  serves  two  purposes :  it  improves  the  insulation  and  con- 
ducts the  heat  from  the  coils  and  core  to  the  outer  case,  where 
it  is  radiated  to  the  surrounding  air.  Fig.  216  shows  the  coils 


TRANSFORMERS.  293 

and  core  of  a  small  Westinghouse  transformer.  It  will  be  noted 
that  this  transformer  is  of  the  "  shell "  type ;  the  iron  core  sur- 
rounds the  coils  which  project  at  each  end  beyond  the  lamina- 
tions. Fig.  217  shows  the  coils  and  core  of  a  Wagner  trans- 
former, which  is  also  of  the  shell  type.  The  primary  PP  is  wound 
in  four  coils,  the  secondary  ,5\9  in  two  coils,  and  the  whole  trans- 


Fig.  218. 

former  is  designed  for  immersion  in  oil.  By  winding  the  coils 
in  sections  and  sandwiching  them  as  shown,  magnetic  leakage  is 
reduced  to  a  minimum.  Openings  are  left  between  the  coils  and 
core,  so  that  the  oil  can  circulate  as  shown  by  the  small  arrows, 
thus  preventing  undue  heating  of  the  interior  parts.  Fig.  218 
shows  the  core  and  coils  of  a  large  Westinghouse  transformer 


294  ALTERNATING   CURRENT    MACHINERY. 

of  a  type  used  for  power  transmission  work.  It  is  of  375  kilo- 
watts capacity  and  its  primary  is  wound  for  33,000  volts.  The 
flat  primary  and  secondary  coils  are  clearly  shown;  they  are 
flared  out  at  the  end  so  as  to  give  ventilation.  The  arrange- 


Fig.  219.  Fig.  220. 

ment  of  the  core  and  coils  is  practically  the  same  as  that  of  the 
small  transformer  in  Fig.  216,  the  coils  being  surrounded  by  the 
iron. 

Fig.  219  shows  the  arrangement  of  coils  and  core  for  a  Gen- 
eral Electric  Type  H  transformer  of  small  size,  and  Fig.  220 
shows  a  section  through  the  transformer  in  its  case.  As  seen 
in  Fig.  219,  the  transformer  is  of  the  "core"  type,  since  the 
coils  surround  the  laminated  iron  core. 

188.  Cooling  of  transformers. — Transformers  of  moderate  size 
are  capable  of  radiating  the  heat  generated  in  them  without  any 
special  means  being  provided  for  carrying  off  the  heat.  The  su- 
perficial area  of  these  smaller  transformers  is  sufficient  to  radiate 
the  heat  arising  from  the  various  losses.  However,  as  the  out- 
put of  transformers  is  increased,  the  radiating  area  increases  in 
a  much  lower  proportion  than  the  output;  hence,  large  trans- 
formers are  not  capable  of  getting  rid  of  the  heat  generated  in 
them  without  attaining  a  dangerously  high  temperature,  and 


TRANSFORMERS. 


295 


special  means  have  to  be  provided  for  ventilating  them.  The 
fact  that  special  cooling  facilities  must  be  provided  for  large 
substation  transformers  does  not,  in  any  sense,  mean  that  these 
transformers  are  not  efficient.  As  a  matter  of  fact,  a  large  trans- 
former is  one  of  the  most  efficient,  if  not  the  most  efficient, 
piece  of  apparatus  used  for  transforming  energy.  A  large  sub- 
station transformer  may  have  an  efficiency  of  over  98  per  cent, 
at  full  load.  The  heating  does  not,  therefore,  represent  an  ex- 
cessive waste  of  energy ;  it  simply  means  that  the  transformer  has 


Fig.  221. 

such  small  linear  dimensions  compared  with  its  output  that  not 
enough  radiating  surface  is  available  to  get  rid  of  the  heat  with- 
out some  special  provision  for  the  purpose.  It  is  much  more 
economical  to  provide  special  cooling  facilities  than  to  make  the 
transformer  large  enough  to  radiate  the  heat  itself.  Self-cooling 
transformers  of  large  output  are  therefore  higher  in  cost  per 
kilowatt  than  those  designed  for  artificial  cooling. 


296  ALTERNATING    CURRENT   MACHINERY. 

Some  of  the  methods  used  for  cooling  are  as  follows :  Use  of 
oil  to  conduct  the  heat  to  the  outer  case,  oil  circulation,  water 
circulation,  and  air  blast.  In  oil-cooled  transformers  where  the 
heat  is  simply  conducted  to  the  case  by  means  of  the  oil  and 
then  radiated  to  the  surrounding  air,  the  cases  are  usually  pro- 
vided with  deep  corrugations  so  as  to  present  a  large  radiating 
surface. 

Transformers  have,  in  a  few  instances,  been  cooled  by  provid- 
ing an  oil-circulating  pump  to  keep  up  a  circulation  of  oil  around 
the  coils  and  core.  More  often,  however,  the  cooling  is  accom- 
plished by  filling  the  case  with  oil  and  cooling  the  oil  by  circu- 
lating water  through  a  coiled  pipe  placed  near  the  top  of  the 
case.  Fig.  221  shows  a  large  Stanley  transformer  with  the  outer 
case  removed.  This  transformer  is  of  the  oil-immersed  water- 
cooled  type,  the  oil  being  cooled  by  means  of  water  circulating 
in  the  coil  of  pipe  A  mounted  in  the  upper  part  of  the  case.  The 
heated  oil  rises  from  the  bottom  to  the  top,  where  it  is  cooled, 
and  then  descends  to  the  bottom ;  a  continuous  circulation  of  oil 
is  thus  maintained. 

In  the  air-blast  type  of  transformer,  openings  are  provided  be- 
tween the  coils  and  core.  The  transformer  is  set  over  a  chamber, 
into  which  air  is  forced  by  a  fan  driven  by  an  electric  motor; 
the  air  enters  through  an  opening  in  the  bottom  of  the  transfor- 
mer casing,  flows  up  through  the  openings  in  the  core  and  be- 
tween the  coils,  and  passes  out  at  the  top  and  sides.  The  sectional 
view,  Fig.  222,  illustrates  an  air-blast  transformer.  This  method 
of  ventilation  is  clean  and  effective,  and  the  amount  of  power 
required  to  run  the  fan  is  but  a  very  small  fraction  (about  •£$  of  I 
per  cent.)  of  the  output  of  the  transformers  to  be  cooled.  In 
Fig.  222,  B  represents  the  core  and  A  one  of  the  coils.  The  pri- 
mary and  secondary  are  subdivided  into  several  flat  coils  that  are 
thoroughly  insulated 'from  each  other  by  separating  diaphragms. 
The  core  is  built  up  in  such  a  way  as  to  leave  air-ducts  cc  at 
regular  intervals.  The  air  enters  at  the  bottom,  passes  up  be- 
tween the  coils  and  through  the  core  and  out  at  the  top  and 


TRANSFORMERS. 


297 


sides.  The  flow  of  the  air  between  the  coils  is  controlled  by  the 
damper  g  at  the  top,  and  another  damper,  not  shown,  regulates 
the  flow  through  the  core.  The  lower  casing  D  is  provided  with 


A/f  Chamber 


Fig.  222. 

doors  ee  to  allow  access  to  the  secondary  terminals  in  case  they 
cannot  be  reached  from  the  air-chamber  below;  one  of  the  pri- 
mary terminals  is  shown  at  h. 

189.  Transformer  connections  on  two-phase  circuits. — In  Amer- 
ica it  is  customary  to  use  separate  transformers  for  each 
phase  of  a  polyphase  system,  although  three-phase  transfor- 
mers are  now  being  used  to  some  extent.  In  Europe,  poly- 
phase transformers  are  more  widely  used.  In  Figs.  22^-226, 
separate  transformers  are  shown  for  each  phase.  Figs.  223  and 


298 


ALTERNATING   CURRENT    MACHINERY. 


224  show  connections  commonly  used  for  two-phase  systems, 
and  Figs.  225  and  226  those  for  three-phase  systems.  In  all  the 
figures  the  primary  voltage,  for  the  sake  of  illustration,  is  taken 


1000  l/o//s 


Fig.  223 


as  1,000  volts,  and  all  the  transformers  are  supposed  to  have  a 
ratio  of  10  to  I,  i.  e.,  the  primary  coils  have  ten  times  as  many 
turns  as  the  secondaries.  Fig.  223  (a)  (b)  shows  transformers 


IOOO  Vo/ts 

t 

^ 

1000  Vo/ts 

I 

T 

1 

h—  Fuse  •( 

1         1 

3                V 

/OOOVo/fs 


l4l4Vo/fs\ 


\  IOOOVoHs\ 


Fig.  224. 


connected  on  a  two-phase  four-wire  system.  In  (a)  a  trans- 
former is  connected  to  each  phase  and  the  secondaries  supply 
separate  lighting  circuits,  or  in  case  a  motor  is  operated  both 


TRANSFORMERS.  299 

phases  are  run  to  the  motor.  Both  phases  are  independent 
throughout,  and  this  constitutes  the  commonest  scheme  of  con- 
nection used  on  the  two-phase  system.  In  case  a  single  trans- 
former alone  is  needed  for  lighting  or  other  purposes,  it  is  con- 
nected across  one  of  the  phases  and  the  balance  of  the  system 
is  maintained  by  connecting  a  transformer  required  at  some  other 
point  on  the  other  phase.  By  exercising  care  in  the  connecting 
of  scattered  transformers  the  load  on  the  two  phases  can  be  kept 
nearly  balanced.  Fig.  223  (b)  shows  two  transformers  with 
their  primaries  connected  across  the  same  phase,  and  their  sec- 
ondaries in  series  for  operating  lamps  on  the  three-wire  system. 
This  is  in  reality  a  single-phase  arrangement,  as  only  one  phase 
of  the  system  is  used.  The  voltage  across  the  two  outside  wires 
is  200  volts  or  twice  that  between  the  middle  or  neutral  wire  and 
either  of  the  outside  wires.  This  arrangement  is  sometimes  used 
where  a  considerable  number  of  lights  are  to  be  supplied,  or 
where  three-wire  secondary  mains  are  used.  It  would  not  be 
used  where  motors  are  operated.  Fig.  224  (a)  shows  an  ar- 
rangement similar  to  223  (b)  except  that  the  primaries  are  con- 
nected to  the  two  phases.  In  this  case  the  voltage  across  the 
outside  secondary  wires  is  only  141  volts  (iooX  V2)>  instead 
of  200,  because  the  two  pressures  of  100  volts  on  each  side  are 
in  quadrature  with  each  other.  Fig.  224  (b)  shows  the  same 
arrangement  as  (a)  except  that  instead  of  four  separate  wires 
three  wires  are  used  for  the  primary,  as  explained  in  Article  69. 
This  scheme  of  connection  is  not  to  be  recommended  where  lights 
are  operated,  because  the  voltages  on  the  two  sides  are  liable  to 
become  unbalanced  to  a  sufficient  extent  to  affect  the  lamps 
seriously.  If  a  three-wire  secondary  lighting  system  is  to  be  fed, 
the  connections  shown  in  223  (b)  are  much  to  be  preferred. 

190.  Transformer  connections  on  three-phase  circuits. — Owing 
to  the  fact  that  either  the  primaries  or  secondaries  of  transform- 
ers operated  on  a  three-phase  system  may  be  connected  Y  or  A 
(star  or  mesh),  there  are  quite  a  number  of  different  methods  of 


3oo 


ALTERNATING   CURRENT   MACHINERY. 


connection  available.  Figs.  225  and  226  show  four  of  the 
methods  most  commonly  used.  The  connections  shown  in  Fig. 
225  (a)  are  used  more  than  any  of  the  others.  Here,  both 
primaries  and  secondaries  are  delta  (A)  connected.  The  pres- 
sure applied  to  the  primary  of  each  transformer  is  the  same 
as  the  line  voltage,  and  the  secondary  pressure  is,  in  this  case, 
one-tenth  of  the  primary,  since  we  have  assumed  that  the  primary 
coils  have  ten  times  as  many  turns  as  the  secondary.  The  chief 
advantages  of  this  scheme  of  connection  are  that  it  admits  the 
use  of  transformers  having  the  ordinary  ratios  of  transformation 


fOOOVoffs 


IOOO  tolls 


IOOOVo/fs 


~-!OOOV-*\ 


00q»  -°    ^  J^j.  -0    ^  Jft» 


—  ^vk'  —  - 

^-5  77V.  —  * 

r 

i 

L^     ^J 

fr/mary 


Fig.  225. 

without  giving  rise  to  odd  secondary  voltages;  also  in  case  one 
transformer  burns  out  or  in  case  its  primary  fuse  blows,  the  serv- 
ice will  not  be  interrupted.  By  cutting  down  the  local  some- 
what, it  can  be  carried  by  the  two  remaining  transformers. 

In  Fig.  225  (b)  the  transformers  are  shown  with  both  their 
primaries  and  secondaries  Y-connected.  The  voltage  across  the 
primary  of  each  transformer  will  be  equal  to  the  line  voltage  di- 
vided by  V3>  which  in  this  case  gives  about  577  volts.  The 
pressure  across  the  secondary  mains  will  be  equal  to  the  pressure 
of  the  secondary  of  one  transformer  multiplied  by  V3-  Thus, 


TRANSFORMERS. 


301 


the  secondary  voltage  of  each  transformer  in  (b)  will  be  57.7, 
because  the  winding  has  a  ratio  of  10 : 1  and  the  voltage  between 
the  secondary  lines  will  be  57.7  X  V3==  IO°-  This  scheme  of 
connection  requires  the  primaries  and  secondaries  to  be  wound 
for  odd  voltages,  and  if  one  transformer  breaks  down,  the  service 
is  crippled.  It  has  the  advantage,  however,  that  the  voltage 
across  the  primary  coils  is  much  less  than  the  voltage  of  the 
mains,  and,  consequently,  where  very  high  pressure  mains  are 
used,  the  insulation  of  the  primary  coils  is  rendered  less  difficult. 
This  is  also  true  of  the  arrangement  shown  in  Fig.  226  (a), 


1 000  Volts 


lOOOVo/fs 


lOOOVolts  \ 


I      I 


\C.  Voter 
3  Phase 


Fig.  226. 

where  the  primaries  are  Y-connected  and  the  secondaries  A-con- 
nected.  The  connections  shown  in  Fig.  226  (a)  give  an  old 
secondary  voltage  unless  transformers  with  a  special  ratio  of 
transformation  are  used. 

Fig.  226  (b)  shows  the  connections  for  two  transformers  on  a 
three-phase  system,  an  arrangement  sometimes  used  for  the 
operation  of  small  motors;  it  is  not  recommended  for  large 
motors.  It  is  equivalent  to  the  delta  connection  with  one  side 
omitted,  and  if  one  transformer  gives  out  the  service  is  inter- 
rupted. Methods  of  connection  other  than  those  shown  are 


302 


ALTERNATING   CURRENT    MACHINERY. 


possible ;  for  example,  the  primaries  might  be  connected  delta 
and  the  secondaries  Y.  Also,  in  some  cases  where  transformers 
are  used  for  lighting,  a  common  return  wire  is  run  from  the  com- 
mon connection  of  the  secondaries,  so  that  a  return  path  will  be 
provided  in  case  the  system  becomes  unbalanced.  The  connec- 
tions shown  are,  however,  the  ones  most  commonly  used. 

191.  Polyphase  transformers. — Fig.  227  shows  two  arrange- 
ments of  coils  and  core  suitable  for  polyphase  transformers ;  (a) 
is  for  a  three-phase  transformer  and  (b)  for  a  two-phase.  In 
(a)  the  three  sets  of  primary  and  secondary  coils  (one  primary 


Fig.  227. 

and  one  secondary  for  each  phase)  are  wound  on  the  three  lam- 
inated cores  ABC  which  are  of  equal  cross-section  and  con- 
nected across  the  ends  by  common  yokes  as  shown.  The  three 
primary  and  three  secondary  coils  may  be  connected  Y  or  A  in 
the  same  way  as  already  described  for  separate  transformers. 
The  magnetic  fluxes  in  the  three  cores  ABC  follow  the  same  laws 
as  the  currents  in  the  wires  of  a  three-phase  system.  For  ex- 
ample, when  the  flux  in  core  A  is  at  its  maximum,  the  fluxes  in 
B  and  C  are  half  as  great  and  in  the  opposite  direction,  each 
core  acts  alternately  as  the  return  path  for  the  fluxes  in  the 
other  two  cores  and  the  three-phase  transformer  is  therefore 
more  economical  in  material  and  occupies  considerably  less  space 


TRANSFORMERS.  303 

than  three  single-phase  transformers  having  a  combined  output 
equal  to  that  of  the  three-phase  transformer.  The  breadth  a  of 
the  cores  and  yoke  should  be  alike,  as  the  flux  in  each  of  these 
parts  is  of  similar  amount. 

In  the  two-phase  transformer  shown  in  (b)  the  construction 
of  the  core  is  similar  to  that  in  (a)  except  that  the  breadth  of 
the  central  core  must  be  0^/2  =  1.414.0.  The  primaries  and 
secondaries  of  the  two-phases  are  wound  on  cores  A  and  B  and 
the  flux  carried  by  the  central  core  is  the  resultant  of  the  fluxes 
in  A  and  B.  Since  the  fluxes  in  A  and  B  are  equal  and  differ  in 
phase  by  90°,  it  follows  that  the  flux  in  the  central  core  is  V^ 
times  as  great  as  in  the  outside  cores,  and  in  order  to  keep  the 
magnetic  density  the  same,  the  cross-section  must  be  V2"  times 
as  great. 

192.  Constant  current  transformer. — When  series  arc  lamps 
are  to  be  run  on  constant  potential  alternating-current  systems, 
it  is  necessary  to  use  some  device  whereby  the  current  in  the 
lamp  circuit  may  be  kept  constant  irrespective  of  the  number  of 
lamps  in  operation.  One  way  in  which  this  can  be  accomplished 
is  by  the  use  of  a  constant-current  transformer,  i.  e.,  a  trans- 
former which,  when  its  primary  is  supplied  with  current  at  con- 
stant potential,  will  deliver  a  constant  secondary  current.  The 
General  Electric  constant-current  transformer  is  one  that  is 
largely  used  for  arc  lighting.  The  principle  of  operation  of 
this  transformer  was  explained  in  Article  118,  and  Fig.  228 
shows  the  working  parts  of  one  of  the  larger  sizes.  The  core  is 
of  the  same  shape  as  that  shown  in  Fig.  143,  but  in  this  case 
there  are  two  movable  coils  and  two  fixed  coils.  The  fixed  coils 
are  placed  at  the  top  and  bottom  as  shown.  The  movable  coils 
are  counterbalanced  against  each  other  by  means  of  the  levers 
and  move  up  and  down  between  the  fixed  coils.  When  the 
transformer  is  fully  loaded,  the  flat  movable  coils  rest  on  the 
fixed  coils,  and  the  transformer  generates  its  maximum  electro- 
motive force.  If  the  load  is  diminished  by  cutting  out  lamps, 


304 


ALTERNATING   CURRENT   MACHINERY. 


the  current  in  the  movable  coils  tends  to  increase,  thus  causing 
a  repulsion  between  the  coils.  When  the  coils  become  separated, 
the  magnetic  leakage  between  them  results  in  a  lowering  of  the 
electromotive  force,  as  explained  in  Article  118.  The  repulsion 


Fig.  228. 

between  the  coils  is  balanced  by  a  small  auxiliary  lever  shown  at 
the  base  of  the  transformer,  and  by  regulating  the  weight  on  this 
lever,  the  current  may  be  adjusted.  '  The  whole  mechanism  is 
placed  in  a  corrugated  iron  case  filled  with  oil,  which  helps  to 
conduct  the  heat  off,  and  also  steadies  the  movements  of  the  coils. 


CHAPTER   XVIII. 
INDUCTION    MOTORS. 

193.  Construction, — In  most  of  the  induction  motors  now  built, 
the  part  into  which  the  currents  are  led  from  the  line  is  the  sta- 
tionary member  or  stator.  That  in  which  the  induced  currents 
are  set  up  is  the  rotating  member  or  rotor.  The  latter  is  also 


Fig.  229. 

called  the  armature,  or  secondary,  and  the  former  the  field,  or 
primary.  The  construction  of  an  induction  motor  is,  on  the 
whole,  comparatively  simple,  especially  if  the  squirrel-cage  type 
of  armature  is  used.  Fig.  229  shows  the  field,  or  primary,  of  a 

305 


306  ALTERNATING    CURRENT    MACHINERY. 

Westinghouse  I5o-kilowatt  motor.  The  winding,  which  is  of 
copper  bars  in  this  case,  is  uniformly  distributed  in  slots,  and  is 
connected  up  in  the  same  way  as  the  windings  described  for  poly- 
phase alternator  armatures.  Fig.  230  shows  the  armature  or 
secondary.  Each  slot  is  provided  with  a  single  rectangular  bar 
and  the  ends  of  the  bars  are  all  connected  together  by  means  of 
short-circuiting  rings  at  each  end  of  the  armature.  Fig.  231 
shows  a  General  Electric  motor.  The  general  arrangement  of 
the  parts  of  an  induction  motor  with  squirrel-cage  armature  will 


Fig.  230. 

be  understood  by  referring  ,to  Fig.  232.  :.A  is  the  cast-iron  frame 
that  supports  the  field  stampings  B.  The  field  coils  C  are  gen- 
erally arranged  in  two  layers  and  are  held  in  place  "by  strips  of 
wood  d  which  engage  with  notches  in  the  teeth.  The  armature 
stampings  e  are  supported  by  the  arms  /  of  the  armature  spider. 
•The  armature  slots  are  closed  with  the  exception  of  a  small  slit 
g,  and  the  bars  h  are  pushed  in  from  the  end.  In  many  European 
motors,  slots  which  are  entirely  closed  are  used,  but  this  makes 
the  stator  coils  more  difficult  to  place  in  position,  although  it  may, 
in  some  respects,  improve  the  performance  of  the  motor.  K  is 


INDUCTION    MOTORS. 


307 


the  short-circuiting  end  ring  connecting  the  end  of  all  the  bars, 
thus  forming  the  squirrel-cage  winding. 


Fig.  231 


Fig.  232. 


308  ALTERNATING    CURRENT    MACHINERY. 

A  motor  with  a  squirrel-cage  armature,  while  it  is  very  simple 
as  regards  its  construction,  is  not  as  desirable  for  some  kinds  of 
work  as  one  with  a  resistance  in  the  armature.  For  a  large 
torque  at  low  speed  or  at  starting  the  low  resistance  squirrel-cage 
motor  takes  more  current  from  the  line  than  one  having  a  start- 
ing resistance  in  the  armature  circuit.  An  induction  motor  with 
high  armature  resistance  will  give  a  good  torque  at  starting  but 
the  high  resistance  will  result  in  reduced  efficiency  and  poor 
speed  regulation  if  it  is  left  in  permanently  while  the  motor  is 
running.  On  the  other  hand,  a  motor  with  a  low  resistance 
squirrel-cage  armature  will  give  good  speed  regulation  and  a 
good  torque  when  running,  but  will  not  give  as  good  a  torque  at 
starting.  For  cases  where  a  large  starting  effort  is  required 


Fig.  233. 

without -taking  an  excessive  current,  the  motor  with  a  resistance 
inserted  in  the  armature  at  starting  is  to  be  preferred.  Of 
course,  squirrel-cage  armatures  can  be  made  of  fairly  high  resist- 
ance, but  this  resistance  is  in  circuit  permanently  and  reduces  the 
efficiency  somewhat,  besides  making  the  speed  regulation  poor. 
Large  starting  currents  are  especially  objectionable  on  circuits 
where  motors  as  well  as  lights  are  operated. 

Fig.  233  shows  an  armature  of  a  General  Electric  motor  which 
carries  a  resistance  within  the  armature  spider.  The  armature 
is  provided  with  a  regular  Y-connected  bar  winding  in  place  of 
the  simple  squirrel-cage.  A  resistance  is  mounted  within  the 


INDUCTION    MOTORS.  309 

armature  and  connected  to  a  sliding  switch  so  that,  by  pushing 
in  the  knob  (at  the  right-hand  end  of  the  shaft  in  Fig.  233),  the 
resistance  may  be  cut  out.  The  resistance  is  divided  into  three 
parts — one  section  in  series  with  each  phase  of  the  armature 
winding.  When  the  switch  is  pushed  in  the  resistance  is  cut  out, 
and  the  three  phases  of  the  winding  are  short-circuited.  By  using 
an  armature  of  this  type,  a  large  starting  torque  may  be  obtained 
without  an  excessive  rush  of  current.  No  collector  rings  are 
necessary  in  connection  with  the  armature,  but  the  construction 


Fig.  234. 

is  considerably  more  complicated  and  expensive  than  that  of  a 
squirrel-cage  armature,  and  the  squirrel-cage  type  is  very  largely 
used  on  account  of  its  simplicity  and  small  amount  of  attention 
required. 

194.  Starting  compensator. — The  rush  of  current  with  a  squir- 
rel-cage motor  can  be  avoided  by  inserting  a  starting  resistance 
in  series  with  the  field  windings  or  by  using  other  means  for 
cutting  down  the  applied  electromotive  force.  The  torque  of  an 
induction  motor  decreases  as  the  square  of  the  applied  voltage, 


ALTERNATING    CURRENT   MACHINERY. 


so  that  this  method  of  starting  results  in  a  greatly  reduced  start- 
ing effort.  However,  in  a  great  many  cases  motors  are  not 
started  up  under  full  load,  so  that  this  is  not  a  serious  objection ; 
whereas,  a  large  rush  of  current  would  be  objectionable  because 
of  the  disturbance  of  other 'parts  of  the  system. 

While  a  resistance  could  be  used  as  described  above,  it  is  more 
economical  to  use  an  autotransformer  (i.  e.,  a  transformer  having 
but  one  coil,  which  serves  both  as  primary  and  secondary)  or 
compensator  as  it  is  usually  called  in  this  connection.  Fig.  234 
shows  a  General  Electric  starting  compensator,  in  which  the 
starting  switch  is  of  the  oil-break  type.  The  Westinghouse  Com- 


Fig.  235. 

pany  also  use  an  arrangement  involving  the  same  general  fea- 
tures. Fig.  235  shows  one  arrangement  of  compensator  connec- 
tions for  a  three-phase  motor.  The  compensator  consists  of  coils 
a,  b,  c,  wound  on  a  laminated  iron  core,  each  coil  being  provided 
with  a  number  of  taps  I,  2,  3,  4.  The  pressure  applied  to  the 
motor  at  starting  will  depend  upon  the  amount  of  the  coil  in- 
cluded in  the  circuit.  For  example,  in  Fig.  235  the  use  of  tap  4 
will  give  a  higher  applied  electromotive  force  than  tap  3,  so 
that  the  starting  current  may  be  adapted  to  the  kind  of  work 
which  the  motor  is  called  on  to  perform. 


INDUCTION   MOTORS.  311 

For  light  work  a  small  voltage  can  be  applied,  in  which  case 
the  current  taken  from  the  line  will  be  small,  while  for  heavy 
work,  where  a  stronger  starting  effort  is  necessary,  a  higher 
electromotoive  force,  with  corresponding  larger  torque,  can  be 
used.  When  a  five-blade  double-throw  switch  is  in  the  lower 
position  shown  in  the  figure,  a  part  of  the  compensator  coil  is  in 
series  with  each  line  leading  to  the  motor,  and  the  voltage  ap- 
plied to  the  motor  is  cut  down.  When  the  switch  is  thrown  to 
the  upper  or  running  position  after  the  motor  has  come  up  to 
speed,  the  motor  terminals  are  connected  directly  to  the  line  and 
the  compensator  windings  are  cut  out,  so  that  the  motor  runs 
under  full  voltage.  The  compensator,  therefore,  prevents  a  rush 
of  current  and  gives  the  motor  a  smooth  start,  although  it  cuts 
down  the  starting  torque  from  what  would  be  obtained  if  the  full 
voltage  were  applied.  In  some  cases  starting  compensators  are 
provided  with  a  controlling  switch  by  which  the  coils  can  be 
cut  out  gradually,  but  for  most  cases  a  simple  double-throw 
switch  which  cuts  all  the  coils  out  at  once  is  sufficient  to  give  a 
smooth  start.  Starting  compensators  of  almost  exactly  the  same 
construction  as  the  above  may  also  be  used  for  starting  synchro- 
nous motors  and  rotary  converters  when  these  machines  are 
started  by  means  of  current  supplied A from  the  alternating  current 
mains.  Small  induction  motors  are  usually  started  by  simply 
closing  the  switch  that  connects  them  to  the  line.  One  advan- 
tage in  using  the  compensator  method  of  starting  is  that  the 
motor  can  be  controlled  from  a  distant  point. 

195.  Speed  regulation  of  induction  motors. — For  some  classes 
of  work  it  is  desirable  to  have  induction  motors  arranged  so 
that  their  speed  can  be  controlled.  The  speed  can  be  regulated  by 
the  insertion  of  an  adjustable  resistance  in  either  the  primary 
or  secondary,  but  the  latter  is  by  far  the  better;  a  reduction 
in  -primary  voltage  greatly  reduces  the  torque  of  the  motor, 
whereas  resistance  in  the  secondary  gives  a  large  torque  at  low 
speeds.  These  methods  may  be  used  when  the  range  of  con- 


312  ALTERNATING    CURRENT   MACHINERY. 

trol  is  not  very  wide.  Generally  speaking  the  induction  motor 
does  not  admit  of  the  same  range  of  control  as  a  direct  current 
motor.  Another  method  that  has  been  used  to  a  limited  extent 
is  to  have  the  winding  on  the  field  arranged  so  that  by  means  of 
a  suitable  controlling  switch,  the  number  of  poles  on  the  motor 
can  be  changed.  Evidently,  the  fewer  the  number  of  poles  the 
faster  a  motor  must  run  when  supplied  with  current  of  a  given 
frequency.  This  is  an  economical  method  of  control,  and  can 
be  used  when  a  wide  change  in  speed  is  desired,  but  introduces 
considerable  complication.  Another  method  that  has  been  used 
for  electric  traction  work  is  applicable  where  two  motors  are 
employed.  This  consists  in  connecting  the  motors  as  follows, 
when  a  low  speed  is  desired.  The  field  of  the  first  motor  is  con- 
nected to  line,  and  the  field  of  the  second  motor  is  supplied  with 
current  from  the  armature  of  the  first  motor.  The  armature  of 
the  second  motor  is  short-circuited.  When  the  motors  are  thus 
connected,  they  will  run  at  half  speed.  When  a  high  speed  is 
desired,  the  motors  are  connected  in  parallel  directly  across  the 
line.  Intermediate  speeds  can  be  obtained  by  combining  the 
use  of  resistances  with  the  above  method;  on  the  whole,  the 
method  is  somewhat  analogous  to  the  series-parallel  control  used 
with  direct-current  street-car  motors. 

Of  the  first  two  methods  of  speed  regulation,  the  one  nearly 
always  used  is  that  by  which  an  adjustable  resistance  is  inserted 
in  the  armature.  It  requires  the  use  of  collector  rings  on  the 
motor,  since  a  resistance  designed  for  continuous  use  for  speed- 
regulating  purposes  is  too  bulky  to  be  placed  within  the  arma- 
ture and  would  moreover  lead  to  too  much  heating  in  the  machine. 
Fig.  236  (a)  shows  a  motor  with  a  rheostat  arranged  for  speed 
control.  The  armature  of  the  motor  is  provided  with  a  regular 
three-phase  Y  winding,  and  the  terminals  are  brought  to  the 
collector  rings  a.  The  rheostat  R  contains  three  resistances 
divided  up  into  sections  that  can  be  gradually  cut  out  by  moving 
the  three-legged  arm  b.  Fig.  236  (b)  shows  the  connections 
diagrammatically ;  I,  2,  3  are  the  three  phases  of  the  winding 


INDUCTION   MOTORS. 


313 


on  the  armature,  and  rlf  r2,  rz  the  three  sections  of  the  resistance. 
When  the  motor  is  running  at  its  slowest  speed,  the  windings  are 
connected  together,  with  all  the  resistance  in  series,  by  means  of 
the  arm  b  which  is  represented  in  (b)  by  means  of  the  circles. 
As  the  arm  b  is  moved  over  the  resistance  contacts,  rlt  r2,  and  rz 
are  cut  out  and  when  the  motor  is  running  at  full  speed  the  arm 


Fig.  236. 

short-circuits  the  windings,  as  indicated  by  the  full  line  circle  in 
(b).  The  wires  from  the  motor  field  may  lead  directly  to  the  line 
through  the  main  switch  or,  as  is  frequently  done,  they  may  be 
brought  through  the  rheostat  and  the  arm  b  made  to  answer  both 
as  a  starting  and  regulating  switch  by  letting  current  into  the 
field  when  it  is  placed  on  the  first  position. 

When  fairly  large  motors  are  used,  as  on  hoists  or  pumps,  it 
is  customary  to  have  the  regulating  resistance  entirely  separate 
from  the  controlling  switch.  In  this  case,  a  controller  somewhat 
similar  to  that  used  on  street-cars  is  employed,  and  is  connected 
to  the  resistance  and  to  the  motor  by  means  of  cables,  thus  allow- 
ing the  controller  and  resistance  to  be  placed  in  whatever  location 
mav  be  most  convenient. 


3 14  ALTERNATING   CURRENT   MACHINERY. 

196.  Voltage. — Induction  motors  are  usually  wound  for  no, 
220,  440,  or  550  volts,  though  in  some  cases  they  are  wound  for 
high  pressures  so  as  to  take  current  directly  from  the  high-pres- 
sure mains  without  the  intervention  of  transformers.      Since  the 
field  is  stationary,  it  is  capable  of  being  insulated  for  high  pres- 
sures.     The  majority  of  the  motors  in  common  use  are,  how- 
ever,  operated   at   the   above   voltages    and    are    supplied    from 
transformers.      It  is  important  that  the  voltage  supplied  to  an 
induction  motor  be  kept  up  to  its  normal  amount.     The  output 
of  the  motor  decreases  as  the  square  of  the  voltage,  and  if  the 
pressure  becomes  low,  the  motor  will  have  very  little  margin  for 
momentary  overloads. 

197.  Current. — The  current  taken  by  an  induction  motor  at 
full  load  will  depend  upon  the  power  factor  and  efficiency  of  the 
motor,  as  well  as  the  output  and  the  voltage  of  supply.     For  a 
three-phase  motor  the  current  in  each  line  will  be 

W 
=  £X/X,Xi.732 

where  W  =  the  watts  output; 

/^ power  factor; 

<?  =  commercial  efficiency. 

The  power  factor  of  induction  motors  of  ordinary  size  and 
frequency  varies  from  .75  to  .90  at  full  load.  Some  of  the 
larger  motors  may  run  a  little  higher  than  this;  .85  is  a  fair 
average  for  motors  of  medium  size.  The  commercial  efficiency 
varies  through  about  the  same  range.  For  a  two-phase,  four- 
wire  motor,  the  current  in  each  line  will  be 

W 
C= (129) 

£X/X?X2 

where  the  quantities  W,  E,  f  are  the  same  as  before.  The  follow- 
ing table  given  by  the  General  Electric  Company  shows  the  cur- 
rent taken  in  each  line  by  three-phase  no-volt  motors.  For 
higher  voltages  the  current  will  be  smaller  in  proportion. 


INDUCTION    MOTORS. 


315 


Horse  Power  of  Motor.         Full  Load  Current. 

Horse  Power  of  Motor. 

Full  Load  Current. 

I 

6.3 

2O 

112 

2 

12 

30 

168 

3 

18 

50 

268 

5 

28 

75 

390 

10 

54 

100 

55o 

16 

81 

150 

780 

198.  Frequencies. — The  common  frequencies  for  which  induc- 
tion motors  are  built  in  America  are  25,  40,  60,   125  and  133 
cycles  per  second.      The  high-frequency  motors   (125  and  133) 
were  never  built   in  very  large   sizes   and  are  now   practically 
obsolete.     The  high  frequency  tends  to  aggravate  the  effects  of 
lagging  currents.     These  effects  may  be  overcome  to  a  consider- 
able extent  by  using  condensers  at  the  motor  to   improve   its 
power  factor,  the  wattless  current  being  supplied  by  the  conden- 
ser;  the    Stanley   Company   at   one   time   used   condensers    for 
this  purpose.     Nearly  all  polyphase  motors^are  now  operated  on 
25  or  60  cycles. 

199.  Transformers  for  induction  motors. — The  size  of  trans- 
formers to  be  used  for  supplying  a  given  induction  motor  will,  of 
course,  depend  upon  the  power  factor  and  efficiency  of  the  motor 
as  well  as  on  its  output.     A  rule  that  is  commonly  followed,  and 
which  is  a  safe  one  at  least  for  motors  of  the  larger  sizes,  is  to 
install  i  kilowatt  of  transformer  capacity  for  every  horse-power 
output  of  the  motor.      For  example,   a   75-horse-power   three- 
phase  motor  would  require  75  kilowatts  in  transformers,  or  three 
transformers  of  25  kilowatts  each.     The  General  Electric  Com- 
pany recommends  the  use  of  transformers  as  indicated  in  the 
following  table: 


Size  of  Motor. 


Kilowatts  of  each  Transformer. 


Horse  Power. 

2  Transformers. 

3  Transformers. 

I 

.6 

.6 

2 

i.5 

i 

3 

2 

1.5 

5 

3 

2 

7^ 

4 

3                • 

10 

6 

4 

15 

7.4 

5 

20 

10 

7-5 

30 

15 

10 

50 

25 

15 

75 

40 

25 

3l6  ALTERNATING    CURRENT   MACHINERY. 

200.  Direction  of  rotation  of  induction  motors. — The  direction 
of  rotation  of  an  induction  motor  is  determined  by  the  direction 
in  which  the  magnetic  field  rotates,  and  this  in  turn  is  determined 
by  the  relation  of  the  currents  in  the  windings  to  each  other. 
The  direction. of  rotation  of  a  three-phase  motor  may  be  changed 
by  interchanging  any  two  of  the  wires  leading  to  the  field  wind- 
ing.    To  reverse  a  two-phase  motor,  the  two  wires  of  one  of  the 
phases  would  be  interchanged. 

201.  Single -phase    induction    motor. — Single-phase    induction 
motors  are  constructed  in  the  same  manner  as  polyphase  motors 
except  that  the  stator  is  provided  with  a  single-phase  winding. 
When  such  a  motor  is  standing  still  and  the  current  turned  on, 
the  magnetic  field  set  up  in  the  air  gap  is  a  simple  oscillating 
field,  i.  e.,  it  does  not  rotate  as 'in  the  polyphase  motor,  and  there 
is  therefore  no  turning  effort  exerted  on  the  rotor.     Such  a  mo- 
tor will  not  start  up  of  its  own  accord,  but  if  the  rotor  is  started 
from  an  outside  source  it  will  soon  run  up  to  speed  and  the 
machine  will  then  behave  very  much  like   a  polyphase   motor. 
When  the  rotor  is  running  near  synchronism  the  secondary  cur- 
rents  induced   therein   are    displaced   in   phase    with   regard   to 
the   stator  currents,   so  that  the   magnetic   field   in   the   air-gap 
revolves   in   much  the   same   manner  as   in   a  polyphase   motor, 
though  the  field  produced  is  not  a  perfect  revolving  field  un- 
less the   rotor   is   running  very   nearly   at   synchronism.      If   a 
single-phase  motor  is  to  exert  a  torque  at  standstill  and  run  up 
to  speed  under  load,  it  must  be  arranged  so  that  a  revolving  field 
can  be  set  up  at  starting.      In  some  cases  this  is  done  as  ex- 
plained in  Art.  154,  the  rotor  being  provided  with  an  auxiliary 
winding  which  is  open-circuited  after  the  motor  has  attained  its 
speed.     In  this  case  the  necessary  phase  displacement  of  the  cur- 
rent in  the  windings  is  obtained  by  using  an  inductance  in  one 
branch.      Fig.  237  shows  an  arrangement  used  by  the  General 
Electric  Company.     The  motor  is  provided  with  three  windings, 
cde,  as  in  a  three-phase  motor,  and  one  of  these  windings  is  sup- 


INDUCTION    MOTORS. 


317 


plied  through  a  condenser-compensator  ba.  This  consists  of  a 
condenser  b  connected  across  a  compensator  or  auto-transformer 
a.  The  object  of  the  auto-transformer  is  to  increase  the  poten- 
tial applied  to  the  condenser  and  permit  the  use  of  a  small  con- 
denser worked  at  high  potential  rather  than  a  large  condenser 
worked  at  low  potential.  The  impedance  due  to  the  condenser- 
compensator  is  such  that  at  starting  a  leading  current  flows  in 
winding  c.  This  gives  an  imperfect  revolving  field  sufficient  to 
start  the  motor  with  considerable  torque.  The  condenser  has 


Fig.  237. 


Fig.  238. 


the  advantage  that  it  neutralizes  the  inductance  of  the  motor  and 
insures  a  power  factor  of  nearly  unity  at  all  loads.  For  this 
reason  the  condenser-compensator  is  left  in  circuit  even  after 
the  motor  has  been  started. 

202.  Repulsion  motor. — If  an  ordinary  direct-current  arma- 
ture, provided  with  a  commutator,  be  placed  between  pole  pieces 
AB,  Fig.  238,  which  are  excited  by  means  of  single-phase  alter- 
nating current,  and  if  brushes  de  which  are  connected  together, 


318  ALTERNATING    CURRENT    MACHINERY. 

be  placed  at  an  angle  a  with  the  center  line  of  the  poles,  a  consid- 
erable torque  will  be  exerted  on  the  armature  and  it  will  run  up 
to  speed.  If  the  brushes  were  placed  in  the  vertical  position  the 
E.M.F.  between  the  two  brushes  would  be  a  maximum  and  the 
current  flowing  in  the  armature  would  also  be  a  maximum.  If 
the  brushes  were  placed  in  the  horizontal  position  (a  =  90°) 
the  E.M.F.  between  the  brushes  would  be  zero  or  in  general  the 
E.M.F.  would  be  E  cos  a  where  E  is  the  E.M.F.  between  the 
brushes  when  they  are  in  the  vertical  position.  The  current  would 
then  be 

Ecosa 
~^~' 

where  Z  is  the  impedance  of  the  armature.  If,  on  the  other 
hand,  a  given  current  /  is  made  to  flow  in  the  armature,  the 
torque  with  the  brushes  in  the  vertical  position  (a  =  o)  would 
be  zero,  while  with  the  brushes  in  the  horizontal  position 
(a  —  90°)  the  torque  would  be  a  maximum;  in  general,  the 
torque  would  be 


where  K'  is  a  constant  and  3>  the  flux  threading  the  armature. 
However  it  has  just  been  shown  that  for  a  given  angle  a  the 
current  is 

E  cos  a 

~Z^~' 
hence 

T  —  K'®  —  -^  —  .  sin  a  =  K"  sin  a  cos  a  =  K"  sin  2a 

where  K"  is  a  constant  =K'$E/Z.  Since  T  =  J£"sin2a,  the 
torque  theoretically  becomes  a  maximum  where  a  =  45°.  How- 
ever the  above  equations  take  no  account  of  armature  reaction  or 
other  influences  that  affect  the  torque  and  in  motors  as  actually 
built  the  maximum  torque  is  obtained  for  a  value  of  a  much  less 
than  45°.  A  motor  constructed  in  this  manner  will,  therefore, 
operate  on  a  single-phase  current  and  will  start  with  a  good  torque 


INDUCTION   MOTORS. 


319 


but  it  necessitates  the  use  of  a  commutator  with  its  attending 
troubles  and  for  this  reason  it  has  been  used  comparatively  little. 
The  principle  of  the  repulsion  motor  is,  however,  utilized  in  the 
Wagner  single-phase  motors  for  securing  a  good  starting  effort.* 

203.  Wagner  single-phase  motor. — Fig.  239  shows  the  con- 
struction of  a  successful  type  of  single-phase  motor  made  by  the 
Wagner  Electric  Mfg.  Co.  The  stator  A  A'  is  provided  with  a 
single-phase  winding  placed  in -slots  in  a  laminated  structure  in 
the  usual  manner.  The  armature  or  rotor  BB  is  provided  with 


Fig.  239. 

a  regular  direct-current  drum  winding  connected  to  the  segments 
cc  of  a  radial  commutator.  Short-circuited  brushes,  one  of 
which  is  shown  at  e,  are  arranged  to  press  on  the  commutator. 
During  the  starting  period  the  brushes  press  on  the  commutator 
and  the  machine  starts  up  as  a  repulsion  motor  with  a  good 
torque.  When  speed  has  been  attained,  the  weights  ff  fly  out 

*  For  a  complete  discussion  of  the  repulsion  motor  see  Chap.  XX  Stein- 
metz's  Alternating  Current  Phenomena. 


320 


ALTERNATING    CURRENT    MACHINERY. 


on  account  of  centrifugal  force,  and  push  the  short-circuiting 
ring  dd  to  the  left,  thus  connecting  all  the  inner  edges  of  the 
commutator  bars  together  and  short-circuiting  all  the  armature 
coils.  At  the  same  time  the  brushes  are  pushed  away  from  the 
commutator.  During  regular  operation,  therefore,  the  machine 
is  an  induction  motor  with  a  short-circuited  armature  and  oper- 
ates in  the  same  way  as  one  having  an  armature  of  the  short- 
circuited  type. 

Fig.  240*  shows  the  relation  between  the  angle  a,  Fig.  238, 
and  the  torque  for  a  two-pole  motor  with  commutator-starting 
device.  It  will  be  noticed  that  the  maximum  torque  is  obtained 


16 
14 

,  —  ' 

—  „ 

/ 

^ 

\ 

/ 

X 

8 
6 

2 

/ 

^ 

X, 

/ 

x 

^. 

1 

5 

*^^ 

*•—  . 

1 

1 

2     4      6     8    10  12  14  IB   18   2O  22  24  28  28  SO  82  34 

Va/ue£  offJiDegree-s 

Fig.  240. 

for  a  value  of  a— 10°  instead  of  45°,  as  in  Art.  202,  where  the 
formulas  neglect  armature  reaction  and  assume  that  the  field 
magnet  is  arranged  as  in  Fig.  238  instead  of  completely  sur- 
rounding the  armature  and  having  a  winding  distributed  in  slots, 
as  is  always  the  case  in  the  motors  as  actually  built. 

204.    Use  of  alternating  current  for  short-distance  transmission 

— The  simplicity  of  the  induction  motor  has  been  largely  account- 
able for  the  extensive  use  of  alternating  current  in  connection 
with  short-distance  transmission  plants.  While  alternating  cur- 
rent is  preeminently  adapted  for  the  transmission  of  power  over 
long  distances  because  of  the  ease  with  which  it  admits  the  use 


*  Edw.  Bretch,  American  Electrician,  Vol.  IV,  No.  8. 


INDUCTION    MOTORS.  321 

of  high  pressure,  yet  its  use  at  comparatively  low  pressure  for 
short-distance  transmission,  without  the  intervention  of  trans- 
formers, is  by  no  means  limited.  Some  very  large  plants  have 
been  installed  in  connection  with  various  manufacturing  concerns 
where  polyphase  alternating-current  transmission  has  superseded 
the  use  of  long  lines  of  shafting,  thus  resulting  in  a  large  economy 
of  power  and  allowing  extensions  to  be  readily  made.  Where 
the  machinery  is  started  and  stopped  frequently,  and  where  a  large 
range  of  torque  and  speed  is  desired,  the  direct-current  motor 
probably  gives  the  best  results,  but  for  other  classes  of  work  the 
induction  motor  has  many  advantages  and  has  been  installed  in 
preference  to  the  direct-current  motor  for  many  kinds  of  work 
connected  with  factories.  The  principal  advantage  is  the  absence 
of  the  commutator  and  sliding  contacts,  thus  allowing  the  motor 
to  be  operated  in  places  where  a  direct-current  motor  would  give 
trouble.  For  example,  induction  motors  have  found  extensive 
use  in  cotton  mills  and  in  many  places  where  the  sparking  of  a 
direct-current  machine  would  be  a  source  of  danger. 


CHAPTER   XIX. 
SYNCHRONOUS    MOTORS. 

205.  Synchronous  motors  are  used  for  work  where  power  is 
required  in  comparatively  large  amounts,  and  where  the  motor 
is   not   started   or  stopped   frequently.      A   synchronous   motor 
always  runs  at  the  same  frequency  as  the  alternator  that  supplies 
it  with  current,  and  its  speed  cannot  change  unless  the  speed  of 
the  generator  changes.      The  speed  of  the  motor  may  or  may 
not  be  the  same  as  that  of  the  generator.      It  will  be  the  same 
only  when  the  two  have  the  same  number  of  poles.    The  synchro- 
nous motor  cannot  be  used  where  a  variable  speed  is  required, 
and  it  is  not  suitable  for  work  where  the  motor  has  to  start  up 
under  a  load. 

206.  Construction  of  synchronous  motors. — Synchronous  motors 
are  the  same,  as  regards  their  construction,  as  alternators.     Fig. 
241  shows  a  two-phase  synchronous  motor  of  the  revolving  field 
type.     The  small  motor  is  used  for  starting  purposes,  as  will  be 
explained  later.     The  field  of  the  motor  is  excited  by  a  small 
direct-current  exciter  either  direct-driven  or  belted  to  the  motor. 
Synchronous  motors  operate  best  on  fairly  low  frequencies,  most 
of  them  being  designed  for  frequencies  ranging  from  25  to  60 
cycles  per  second.      Single-phase  synchronous  motors  are  now 
seldom  installed,  though  they  were  used  to  some  extent  before 
polyphase  systems  were  introduced. 

207.  Starting  synchronous  motors. — If   a  polyphase   synchro- 
nous motor  is  connected  to  the  line,  it  will  run  up  to  synchronism, 
but  in  so  doing  it  will  take  a  very  large  current.     When  current 
flows  through  the  armature,  it  reacts  on  the  residual  magnetism 

322 


SYNCHRONOUS   MOTORS. 


323 


set  up  in  the  pole  pieces  by  the  current  in  the  preceding  phase 
and  produces  sufficient  torque  to  start  the  motor.  Of  course, 
this  imperfect  motor  action  does  not  produce  sufficient  torque  to 


enable  the  motor  to  start  under  loa'd,  and  the  method  is  objec- 
tionable on  account  of  the  large  lagging  current  which  the  motor 
takes  from  the  line,  and  which  is  likely  to  create  disturbances  on 
other  parts  of  the  system.  This  method  of  starting  is  not,  there- 


324  ALTERNATING    CURRENT   MACHINERY. 

fore,  admissible  unless  the  motor  is  a  small  one.  If  the  synchro- 
nous motor  has  solid  pole  pieces,  the  eddy  currents  induced  in 
them  give  rise  to  considerable  starting  torque  owing  to  the  ma- 
chine acting  as  an  induction  motor — with  laminated  poles,  the 
torque  from  this  source  would  be  small.  A  single-phase  motor 
will  not  start  up  of  its  own  accord  at  all,  because  the  torque  is 
exerted  rapidly,  first  in  one  direction  and  then  in  the  other,  with 
the  result  that  the  armature  remains  at  a  standstill.  Single- 
phase  machines  must  be  brought  up  to  speed  from  some  outside 
source. 

On  account  of  the  large  rush  of  current  and  large  voltage  drop 
caused  by  connecting  the  motor  directly  to  the  line,  polyphase 
synchronous  motors  are  sometimes  started  by  using  a  starting 
compensator.  This  cuts  down  the  voltage  applied  to  the  motor 
at  starting,  and  is  cut  out  after  speed  is  attained.  Where  large 
motors  are  employed,  the  method  of  starting  that  causes  least  dis- 
turbance is  to  use  a  small  auxiliary  induction-motor,  either  geared 
to  or  mounted  on  an  extension  of  the  main  motor-shaft.  The 
motor  shown  in  Fig.  241  is  started  in  this  way.  After  the  large 
motor  has  been  brought  up  to  speed,  the  small  motor  is  discon- 
nected by  means  of  a  clutch.  A  synchronous  motor  will  run  in 
either  direction,  depending  upon  the  direction  in  which  it  is  driven 
up  to  synchronism.  If,  however,  it  is  started  by  means  of  alter- 
nating current  in  the  armature,  its  direction  of  rotation  is  deter- 
mined by  the  rotating  field  set  up  by  the  armature  windings. 

Synchronous  motors  have  the  disadvantage  of  being  incapable 
of  starting  under  load,  and  of  requiring  considerable  auxiliary 
apparatus  in  the  shape  of  an  exciter  and  starting  motor  or  start- 
ing compensator.  On  the  other  hand,  they  have  some  advan- 
tages. They  are  a  good  motor  for  work  where  a  constant  speed 
is  desirable,  because  so  long  as  the  speed  of  the  generator  does 
not  change,  the  speed  of  the  motor  will  remain  constant.  The 
drop  in  the  line,  therefore,  does  not  cause  the  speed  to  fall  off. 

Another  valuable  feature  of  the  synchronous  motor  is  that  its 
power  factor  can  be  controlled  by  varying  the  field  excitation  so 


SYNCHRONOUS  MOTORS. 


325 


that,  with  care  in  adjusting  the  field  strength,  the  power  factor 
can  be  kept  at  or  near  unity.  Moreover,  if  the  field  is  over-ex- 
cited, the  motor  acts  like  a  condenser  of  large  capacity  and  can 
be  used  to  neutralize  the  lagging  currents  that  may  be  set  up  by 
induction  motors  or  other  pieces  of  apparatus  on  the  same  system. 
The  efficiencies  of  synchronous  motors  are  the  same  as  those 
alternators  of  corresponding  size. 

208.    Prevention  of  hunting. — Synchronous  motors,  like  rotary 
converters,  are  liable  to  hunting  (Art.  129)  and,  to  prevent  this, 


Fig.  242. 

they  are  often  provided  with  dampers  (Art.  142)  to  prevent  the 
oscillations.  These  dampers  are  either  in  the  form  of  closed  cir- 
cuits of  low  resistance  bedded  in  the  pole  face  or  of  low  resistance 
castings  fixed  between  the  pole  tips.  Fig.  242  shows  a  portion 
of  the  revolving*  field  of  a  Bullock  synchronous  motor.  The 
dampers  A  consist  of  castings  held  in  grooves  in  the  edges  of 
the  poles  and  forming  closed  circuits  of  low  resistance.  Any 
momentary  fluctuation  in  speed,  causing  a  cutting  of  the  damper 
by  the  magnetic  flux,  sets  up  powerful  eddy  currents  in  the  dam- 
per and  thereby  checks  the  tendency  to  hunt. 


CHAPTER  XX. 

ROTARY  CONVERTERS  AND  MOTOR  GENERATORS. 

209.  General  construction  of  rotary  converters. — Two  methods 
are  in  general  use  for  changing  alternating  current  to  direct  cur- 
rent, or  vice  versa.  These  are  by  means  of  rotary  converters  and 
motor-generators.  The  principle  of  the  rotary  converter  has 
been  given  in  Chapter  XIII.  In  general  appearance  and  con- 
struction rotary  converters  resemble  direct-current  dynamos  very 
closely.  Fig.  243  shows  the  general  arrangement  of  the  various 
parts  of  a  rotary  converter.  AA  is  the  armature  spider  support- 
ing the  laminated  core  B  which  carries  the  conductors  c  in  slots 
on  its  periphery.  These  conductors  are  usually  in  the  form  of 
copper  bars  and  project  out  at  the  end,  so  that  the  end  connections 
between  the  various  bars  and  to  the  commutator  can  be  readily 
made.  The  projecting  bars  are  supported  by  means  of  the  end 
flanges  fg.  H  is  the  commutator  from  which  or  to  which  the 
direct  current  is  supplied;  k  is  one  of  the  commutator  bars  to 
which  the  armature  winding  is  attached.  L  shows  the  collector 
rings  by  means  of  which  the  alternating  current  is  supplied  to  the 
machine  or  led  from  it  in  case  the  rotary  is  used  to  change  direct 
current  to  alternating.  These  rings  are  connected  to  equidistant 
points  of  the  winding  as  described  in  Chapter  XIII.,  the  number 
of  tapping-in  points  depending  upon  the  number  of  poles  and  the 
number  of  phases  for  which  the  converter  is  intended.  P  is  one  of 
the  pole  pieces  provided  with  a  laminated  pole-shoe  s  and  mag- 
netizing coil  M.  No  pulley  is  necessary,  as  these  machines  simply 
change  the  current  and  are  not  used  as  a  source  of  power  or 
-driven  mechanically. 

326 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  327 

Rotary  converters  are  usually  either  shunt-  or  compound- 
wound,  the  latter  being  used  mostly  for  street  railway  work. 
The  shunt-wound  converter  is  more  stable  in  its  action  than  the 
compound-wound  machine  and  in  many  cases  is  to  be  preferred 
especially  where  the  load  does  not  vary  suddenly.  Rotary 
converters  may  also  be  separately  excited,  but  this  is  not  cus- 
tomary because  the  machine  can  supply  its  own  field  with  cur- 


Fig.  243. 

rent  from  its  commutator  end.  In  the  great  majority  of  cases 
rotary  converters  are  used  to  change  alternating  current  to  direct 
current,  though  occasionally  they  are  used  to  change  direct  cur- 
rent to  alternating.  When  so  used  they  are  sometimes  called 
inverted  rotaries;  a  somewhat  unnecessary  term,  because  there  is 
no  new  feature  in  the  machine,  the  only  difference  being  in  the 
manner  in  which  it  is  used. 


328  ALTERNATING    CURRENT   MACHINERY. 

210.  Starting  of  rotary  converters. — The  rotary  converter, 
when  supplied  with  current  at  the  alternating-current  end,  oper- 
ates essentially  as  a  synchronous  motor;  hence,  what  was  said 
about  the  starting  of  synchronous  motors  applies  in  large  meas- 
ure to  the  starting  of  rotary  converters.  Since,  however,  a  rotary 
converter  combines  the  features  of  a  direct-current  machine  it 
can  also  be  started  by  supplying  current  to  the  direct-current 
side.  A  number  of  different  methods  are  in  use  for  starting 
rotary  converters,  of  which  the  following  are  some  of  the  more 
common  ones. 

(a)  By  connecting  the  alternating-current  side  directly  to  the 
source  of  current.      The  objections  to  this  method  are  the  same 
as  those  given  in  connection  with  the  starting  of  synchronous 
motors.      There  is  a  rush  of  current  which,  on  account  of  its 
large   volume,   gives   rise   to   a   large   drop   in   the   line   and   is 
objectionable  in  its  effects  on  other  parts  of  the  system.      This 
method    of    starting    would    only    be    permissible    with    small 
machines. 

(b)  By    supplying    current    to    the    alternating-current    side 
through  the  medium  of  a  starting  compensator.      This  is  prac- 
tically the  same  method  as  described  in  connection  with  induction 
and  synchronous  motors.     The  autotransformer  supplies  a  large 
current  at  a  low  potential  to  the  machine  and  takes  a  compara- 
tively small  current  at  high  potential  from  the  line,  thus  cutting 
down  the  line  current  and  causing  comparatively  little  disturb- 
ance.    This  method  in  a  somewhat  modified  form  is  used  by  the 
General  Electric  Company  for  starting  low  frequency  converters. 
Instead  of  using  an  autotransformer,  the  regular  transformer 
secondaries  that  supply  the  converter  are  tapped  at  their  middle 
point,  and  by  means  of  a  double  throw  switch  the  voltage  applied 
to  the  converter  at  starting  is  but  half  that  applied  in  regular 
operation.      This  method  will  be  illustrated  more  in  detail  later 
on.     The  chief  advantage  of  this  method  of  starting  is  that  the 
converter  does  not  need  to  be  synchronized  and  hence  can  be  put 
into  operation  quickly. 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  329 


(c)  By  means  of  a  small  auxiliary  induction  motor  mounted 
on  or  geared  to  the  shaft  of  the  converter.  This  arrangement  is 
largely  used  by  the  Westinghouse  Company;  it  is  a  simple 
method  of  starting,  and  one  that  takes  but  a  small  current  from 
the  line.  Fig.  244  shows  a  Westinghouse  rotary  converter  with 
an  induction  motor  A,  arranged  for  starting,  the  armature  of  the 
starting  motor  being  mounted  on  an  extension  of  the  shaft. 


Fig.  244. 

(d)  By  supplying  direct  current  to  the  commutator  end  of  the 
machine  and  starting  the  converter  as  a  direct-current  motor. 
This  method  is  the  one  generally  used  where  direct  current  is 
available  for  starting  purposes.  In  some  cases  the  direct  current 
is  obtained  from  another  converter  already  in  operation  or  from 
a  storage  battery  which  may  be  at  hand.  Sometimes  a  small 
motor-generator  set,  consisting  of  an  induction  motor  coupled  to 
a  direct-current  machine,  is  installed  to  supply  current  for  start- 
ing purposes.  Starting  from  the  direct-current  side  is  preferable 
on  account  of  the  fact  that  when  a  rotary  is  started  from  the 


330  ALTERNATING    CURRENT    MACHINERY. 

alternating-current  side,  the  polarity  of  either  of  the  direct- 
current  terminals  may  be  positive  or  negative ;  that  is,  one  direct- 
current  terminal  may  be  positive  when  the  machine  is  started 
at  one  time,  and  the  next  time  it  is  started  the  same  terminal 
may  be  negative.  Also,  when  either  a  converter  or  synchro- 
nous motor  is  started  by  allowing  alternating  currents  to  flow 
in  the  armature  coils,  the  alternating  magnetism  set  up  through 
the  field  coils  induces  very  high  electromotive  forces  in  them, 
so  that  it  has  been  found  necessary  to  use  special  switches  to 
disconnect  the  field  coils  from  each  other  at  starting.  This  pre- 
vents the  induced  electromotive  forces  from  adding  together  and 
causing  a  breakdown  of  the  field-spool  insulation.  After  the 
machine  has  attained  synchronism,  this  switch  is  closed  and  the 
coils  supplied  with  exciting  current  from  the  direct-current  side. 
When  running  a  rotary  converter  as  a  direct-current  motor  at 
starting,  care  must  be  taken  to  see  that  the  field  circuit  is  intact, 
otherwise  the  armature  may  attain  a  dangerously  high  speed,  be- 
cause under  this  condition  the  machine  no  longer  runs  as  a 
synchronous  motor  but  as  a  direct-current  shunt  motor. 

(e)  By  starting  the  generator  and  motor  together.  This 
method  cannot  often  be  applied  in  practice,  but  where  it  is  possi- 
ble, it  makes  a  good  method  of  starting,  especially  where  large 
units  are  concerned.  The  machines  are  in  this  case  always  in 
phase,  and  the  starting  current  is  comparatively  small. 

211.    Transformation  from  direct  current  to  alternating  current. 

— The  behavior  of  a  rotary  converter  when  used  to  change  from 
direct  current  to  alternating  is  considerably  different  than  when  it 
is  used  in  the  ordinary  way.  When  supplied  with  direct  current, 
it  runs  as  a  direct-current  motor ;  hence,  its  speed  will  vary  with 
changes  in  the  applied  voltage,  and  also  with  changes  in  the  field 
strength.  If  the  field  becomes  weakened,  the  converter  will  speed 
up,  and  a  break  in  the  field-exciting  circuit  may  result  in  damage, 
due  to  racing.  Changing  the  field-excitation  will  not  change  the 
voltage  of  the  alternating  current,  because  the  ratio  of  transfor- 
mation is  fixed  and  changing  the  field  strength  merely  makes  the 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  331 

speed  of  the  converter  change.  The  voltage  of  the  alternating 
current  may  be  changed  by  changing  the  voltage  of  the  applied 
direct  current,  or  it  may  be  increased  or  decreased  by  using 
alternating-current  potential  regulators,  as  described  later. 

Under  certain  conditions,  a  converter  fed  from  the  direct- 
current  side  may  be  subject  to  wide  variations  in  speed,  unless 
special  means  are  taken  to  prevent  it.  For  example,  if  the 
alternating-current  side  delivers  a  large  lagging  current,  the 
armature  exerts  a  powerful  demagnetizing  action  on  the  field, 
which  may  result  in  the  speed  running  dangerously  high.  One 
method  used  by  the  Westinghouse  Company  to  prevent  this  is 
as  follows:  The  converter  is  separately  excited  by  a  small 
direct-current  machine  directly  coupled  like  the  small  induc- 
tion motor  in  Fig.  244.  This  small  shunt-wound  machine  is 
operated  with  an  under-saturated  magnetic  circuit,  so  that  any 
increase  of  speed  will  cause  a  rapid  building  up  of  the  field  and 
a  rapid  increase  in  voltage.  If  the  speed  of  the  rotary  tends  to 
increase,  the  voltage  of  the  exciter  at  once  increases  and 
strengthens  the  field-magnet,  thus  checking  the  tendency  to  race. 
To  accomplish  the  same  protection  the  General  Electric  Company 
places  a  centrifugal  device  on  the  shaft  of  the  converter  which, 
when  the  speed  rises  above  a  certain  amount,  closes  an  electric 
circuit  which  results  in  the  shunting  off  of  the  power  from  the 
converter. 

212.  Transformer  connections  for  rotary  converters. — Rotary 
converters  are  usually  required  to  deliver  current  at  compara- 
tively low  voltage,  generally  700  volts  or  lower.  The  alternating 
current  is,  as  a  rule,  transmitted  at  high  voltage  and  moreover, 
as  shown  in  Article  144,  the  ratio  of  transformation  between  the 
alternating  current  and  direct  current  sides  is  fixed.  For  these 
reasons  it  is  nearly  always  necessary  to  use  transformers  to  step- 
down  the  line  pressure.  Fig.  245  shows  two  three-phase  con- 
verters connected  to  their  transformers.  It  will  be  noted  that  the 
transformers  are  -connected.  This  method  of  connection  is 
generally  preferred  to  the  Y  scheme,  because  in  case  one  of  the 


332 


ALTERNATING   CURRENT    MACHINERY. 


transformers  should  burn  out  or  its  fuse  blow,  the  service  would 
not  be  entirely  crippled.  In  this  figure  T  is  a  small  potential 
transformer  used  to  step-down  the  pressure  for  the  voltmeter  V , 
and  also  to  supply  current  for  the  synchronizing  lamps  /,  /,  or 
synchronism  indicator  as  the  case  may  be.  By  inserting  a  plug 
at  p  or  p'f  the  lamps  will  indicate  when  the  electromotive  force 


I  High  Tens /on  Bus -bars 


Machine  Ml 


\  \ 


D/recf  Cur -re/if   Bus -bars 


Fig.  245. 

of  either  converter  is  in  synchronism  with  the  line  electromotive 
force.  V  and  V"  are  voltmeters  to  indicate  when  the  converter 
is  up  to  voltage.  In  this  figure,  the  transformer  switches  are 
connected  in  the  high-tension  side,  though  in  some  cases  the 
switching  is  done  on  the  low-tension  side  between  the  transformer 
secondaries  and  the  converter.  Switching  on  the  low-tension 
side  is  to  be  preferred  when  the  line  pressures  are  very  high. 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  333 

213.  Connections  for  six-phase  converters. — It  was  shown  in 
Article  143  that  the  output  of  a  converter  is  greater  if  supplied 
with  six  phases  than  if  supplied  with  three  only,  because  the 
armature  heating  is  reduced.  The  use  of  six  phases  necessitates 
six  collector  rings  and  makes  the  connection  somewhat  more 
complicated.  At  the  same  time  the  gain  in  output  is  large 
enough  (40  to  50  per  cent,  over  that  of  a  three-phase  converter) 
to  warrant  the  use  of  six-phase  converters,  especially  where  large 


High  Tension  Bus-bars 


1  High  Tens/on  * 

Switches 


units  are  concerned.  Six-phase  converters  are  extensively  used 
in  connection  with  the  Metropolitan  Street  Railway  System,  New 
York. 

Fig.  249  (a)  shows  transformer  connections  for  supplying  a 
six-phase  converter.  The  current  is  supplied  from  a  three-phase 
high-tension  line  and  transformed  into  six  phases  by  using  a 
double  J  system  of  connections  on  the  secondaries  of  the  trans- 
formers. Each  of  the  transformers  is  provided  with  two  sec- 
ondary coils.  Coils  i,  3  and  5  are  connected  in  the  A  or  mesh 
arrangements,  and  coils  2,  4  and  6  are  also  connected  in  the  same 
way,  except  that  they  are  reversed  with  regard  to  the  first  set. 


334  ALTERNATING   CURRENT   MACHINERY. 

This  gives  the  arrangement  indicated  diagrammatically  in  Fig. 
249  (b).  The  points  a,  b}  c,  d,  e,  f  are  connected  to  the  collector 
rings  of  the  converter  and  from  there  connect  to  equidistant  points 
of  the  armature  winding. 

214.  Frequency  of  rotary  converters. — Rotary   converters  have 
been  most  successful  on  frequencies  ranging  from  25  to  40  cycles. 
They  have  been  built  in  comparatively  large  sizes  for  frequencies 
as  high  as  60  cycles,  but  some  of  these  large  6o-cycle  machines 
have  given  trouble  and  have  been  eventually  displaced  by  motor- 
generator  sets.     Recent  improvements  in  design  have,  however, 
eliminated  many  of  the  defects  formerly  found  in  6o-cycle  con- 
verters.    The  fact  that  a  converter  combines  both  the  features 
of  a  direct  current  generator  and  an  alternating  current  synchro- 
nous  motor  makes   it  difficult  to  construct  large   machines   to 
operate  on  high  frequencies.     With  a  high  frequency,  the  dis- 
tance from  center  to  center  of  the  poles  becomes  so  small  that 
it  is  difficult  to  get  enough  commutator  segments  of  sufficient 
thickness  between  brushes  of  opposite  polarity  without  making 
the  diameter  of  the  commutator  and  armature  very  large,  and 
thus  running  the  peripheral  speed  dangerously  high.     For  these 
reasons,   in  transmission   plants   where  the   bulk  of   the  power 
is  supplied  to  rotary  converters,  it  is  customary  to  install  low- 
frequency  alternators.      A  frequency  of  25  cycles  is  commonly 
used  in  America  for  this  kind  of  work. 

215.  Voltage  regulation  of  rotary  converters. — As   shown   in 
Article  144,  the  ratio  of  transformation  of  a  rotary  converter  is 
practically  a  fixed  quantity.     Assuming  that  the  machine  is  fed 
from  the  alternating-current  side,  changes  in  field  strength  will 
produce   changes    in   the   direct-current   voltage    within   certain 
limits,  if  there  is  considerable  reactance  present  on  the  alternat- 
ing-current side  of  the  rotary.     Strengthening  the  field  of  the  con- 
verter makes  the  current  lead  the  electromotive  force,  and  it  has 
already  been  shown  that  when  a  leading  current  is  delivered  over 
an  inductive  line,  the  voltage  at  the  distant  end  of  the  line  is  in- 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  335 

creased,  and  the  increased  voltage  thus  supplied  to  the  converter 
causes  an  increase  in  the  pressure  of  the  direct-current  side. 
Also,  if  the  field  be  weakened,  the  current  will  lag,  and,  con- 
sequently, the  alternating-current  voltage  will  fall  off,  thus  de- 
creasing the  direct-current  voltage. 

Where  a  comparatively  wide  range  of  voltage  adjustment  is 
needed,  a  number  of  methods  are  available.  One  that  has  been 
used  considerably  is  to  have  the  primary  or  secondary  of  the 
step-down  transformer  divided  into  a  number  of  sections  which 
can  be  cut  in  or  out  by  means  of  a  dial-switch.  This  allows  the 
voltage  applied  to  the  alternating-current  side  of  the  rotary  to 
be  varied  through  a  wide  range,  and  causes  a  corresponding  vari- 
ation in  the  direct-current  voltage. 

Another  method  of  securing  a  considerable  range  of  regulation 
is  to  connect  a  potential  regulator  in  the  lines  running  between  the 
rotary  converters  and  the  transformer  secondaries.  These  poten- 
tial regulators  are  made  in  a  variety  of  forms,  but  they  are  usually 
provided  with  two  windings,  the  primary  of  which  is  connected 
across  the  secondary  of  the  static  transformer,  and  the  secondary 
of  which  is  in  series  between  the  transformer  and  the  converter. 
Whatever  voltage  is  generated  in  this  series  coil  is,  therefore, 
added  to  or  subtracted  from  that  of  the  main  generator,  depend- 
ing upon  the  relation  of  the  electromotive  force  generated  in  the 
series  coil  to  that  generated  in  the  secondary  of  the  main  trans- 
former. Fig.  247  is  a  sectional  view  of  a  General  Electric  three- 
phase  regulator.  This  regulator  is  designed  for  use  with  rotary 
converters,  and  is  fitted  for  hand  control.  Some  of  the  larger 
regulators  are  moved  by  means  of  a  small  auxiliary  motor. 
The  construction  is  similar  to  that  of  an  induction  motor, 
the  secondary  of  which  is  limited  to  small  range  of  move- 
ment. The  primary  winding  ff  is  placed  in  slots  on  the  periphery 
of  a  movable  core  bb  mounted  on  the  shaft  a  in  exactly  the  same 
way  as  the  rotor  winding  of  an  induction  motor.  This  core  can  be 
rotated  through  a  limited  angle  by  means  of  a  worm  g  engaging 
with  the  sector  h ;  current  is  led  into  this  primary  through  flexible 


336 


ALTERNATING    CURRENT    MACHINERY. 


leads.  The  series  winding  ee  is  placed  on  a  stationary  toothed 
core  cc  mounted  in  the  casting  dd.  The  phase  relation  of  the  sec- 
ondary to  that  of  the  primary  (or  the  electromotive  force  of  the 
main  transformer)  can  be  varied  by  moving  the  core,  and,  hence, 
the  pressure  added  to  or  subtracted  from  that  of  the  main  trans- 
former can  easily  be  regulated  by  turning  the  hand  wheel  shown 


Fig.  247. 

in  the  figure.  A  number  of  other  devices  have  been  brought  out 
for  regulating  the  voltage  of  rotary  converters,  but  the  above  will 
give  an  idea  of  some  of  the  more  common  methods. 

216.  Motor  generators. — For  some  kinds  of  work  it  has  been 
found  advantageous  to  use  motor-generator  sets  rather  than  rotary 
converters  for  changing  alternating  current  to  direct.  A  motor 
generator  set  consists  of  an  alternating-current  motor  coupled  to 
one  or  more  direct-current  generators.  One  of  these  sets  is 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  337 

shown  in  Fig.  248.  The  motor  may  be  either  of  the  synchro- 
nous or  induction  type.  The  latter  is  the  less  troublesome  to 
start,  but  the  former  has  the  advantage  that  it  maintains  a  con- 


stant speed  irrespective  of  line  drop,  and,  hence,  has  found  favor 
in  some  lighting  stations.  Motor-generator  sets  are  generally 
more  reliable  than  rotary  converters  on  circuits  operating  at  60 


338 


ALTERNATING    CURRENT   MACHINERY. 


cycles  or  higher.  They  also  admit  of  ready  regulation  of  the 
direct-current  voltage,  because  the  direct-current  machine  is 
entirely  separate  from  the  alternating  current  side  of  the  circuit 
and  its  voltage  can  be  easily  regulated  by  changing  the  shunt 


field  excitation.  Motor-generator  sets  are  used  quite  extensively 
in  connection  with  arc  lighting  where  alternating-current  motors 
are  used  to  drive  arc-light  dynamos.  They  are  also  used  in  some 
places  for  supplying  three-wire  direct-current  systems  from  alter- 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  339 

nating  current.  They  are  somewhat  more  expensive  than  rotary 
converters,  and  are  not  as  efficient.  At  the  same  time  it  must  be 
remembered  that,  when  they  are  used,  the  alternating-current 
motor  can  often  be  wound  so  as  to  take  the  high-pressure  cur- 
rent direct  from  the  line,  thus  doing  away  with  the  static  trans- 
formers, and  the  elimination  of  the  static  transformers  helps 
somewhat  to  even  up  the  difference  in  cost  and  efficiency. 
Moreover,  motor-generators  are  not  so  liable  to  the  peculiar 
effects  met  with  in  connection  with  rotary  converters  such  as 
hunting  and  sparking.  They  are  often  used  in  cases  where 
it  is  desired  to  transform  from  an  alternating  current  of  one 
frequency  to  another  alternating  current  of  higher  or  lower 
frequency,  usually  the  former.  Often  the  current  is  transformed 
from  three-phase  to  two-phase  at  the  same  time.  For  example, 
it  might  be  necessary  to  supply  a  two-phase  high-frequency  light- 
ing system  from  a  low-frequency  three-phase  transmission  system. 
Fig.  249  shows  a  motor  generator  set  used  for  this  work.  It 
consists  of  a  three-phase  revolving-field  synchronous  motor 
coupled  to  a  two-phase  revolving-field  alternator. 

217.  Connections  for  rotary  converter  sub-station — Fig.  250 
shows  a  general  scheme  of  connections  for  a  rotary  converter 
sub-station  where  the  alternating  current  is  received  from  a  high- 
tension  three-phase  transmission  line,  and  after  having  been 
changed  to  direct  current  is  delivered  to  an  electric  railway  system 
at  a  pressure  of  five  or  six  hundred  volts.  Of  course,  the  actual 
connections  used  for  such  sub-stations  vary  considerably,  depend- 

V 

ing,  amongst  other  things,  upon  the  kind  of  apparatus  employed, 
the  purpose  for  which  the  direct  current  is  used,  and  the  method 
of  starting  the  converters.  The  connections  shown  in  Fig.  250 
are  substantially  those  employed  by  The  General  Electric  Com- 
pany for  street  railway  converters  operating  at  twenty-five  cycles 
and  arranged  so  that  they  can  be  started  either  by  direct  or  alter- 
nating current. 

The  incoming  high-tension  line  first  passes  through  a  high- 
tension  switch  in  which  the  circuit  is  broken  under  oil,  thus  effect- 


340 


ALTERNATING   CURRENT    MACHINERY. 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  341 

ually  preventing  arcing  at  the  contacts.  This  switch  allows  all 
current  to  be  cut  off  from  the  sub-station.  The  high-tension  line 
is  connected  to  the  high-tension  bus-bars  from  which  current  is 
supplied  to  the  step-down  transformers.  These  lower  the  voltage 
to  about  350  volts,  since  the  direct  E.M.F.  in  this  case  is  about  550 
volts.  An  oil-break  switch  is  placed  between  the  bus-bars  and  the 
transformers  and  the  primaries  of  the  latter  are  provided  with  a 
number  of  taps,  so  that  they  can  be  adjusted,  within  certain  lim- 
its, to  suit  different  line  voltages.  Both  primaries  and  secondaries 
are  J-connected  and  the  latter  are  connected  to  the  lower  ter- 
minals of  a  double-throw  starting  switch.  The  upper  terminals 
of  this  switch  are  connected  to  the  middle  points  of  the  secondary 
windings,  so  that  when  the  switch  is  thrown  up  during  the  start- 
ing period,  a  reduced  pressure  is  applied  to  the  converter  arma- 
ture. After  the  converter  has  run  up  to  synchronism,  the  starting 
switch  is  thrown  to  the  lower  or  running  position  in  which  full 
pressure  is  applied.  This  arrangement  is,  therefore,  practically 
equivalent  to  a  starting  compensator.  As  the  converter  is  used 
to  furnish  current  to  a  street  railway  system  on  which  the  load 
fluctuates  rapidly,  it  is  necessary  to  provide  an  automatic  means 
for  voltage  regulation.  The  voltage  is,  therefore,  regulated  by 
changing  the  field  strength  of  the  converter  in  combination  \yith 
the  use  of  reactance  between  the  collector  rings  and  the  trans- 
former secondaries.  The  variation  in  field  strength  with  change 
in  load  is  provided  for  by  means  of  the  series  winding  on  the 
converter  field  and  the  reactance  by  means  of  reactance  coils  con- 
nected between  the  starting  switch  and  the  transformer  second- 
aries. It  would,  of  course,  be  out  of  the  question,  with  a  rapidly 
fluctuating  load,  to  use  potential  regulators  for  regulating  the 
voltage  unless  these  regulators  were  in  some  manner  automatically 
controlled.  When  the  converter  is  started  by  connecting  the  col- 
lector rings  to  the  source  of  alternating  current,  it  is  brought  into 
step  by  means  of  the  current  itself  and  it  is  not,  therefore,  neces- 
sary to  go  through  the  process  of  synchronizing  by  means  of 
lamps  or  other  devices.  This  is  often  a  considerable  advantage, 


342  ALTERNATING   CURRENT    MACHINERY. 

especially  when  the  load  is  a  fluctuating  one.  In  order  to  prevent 
high  induced  E.  M.  F.'s  in  the  shunt  field  of  the  converter  due  to 
the  alternating  flux  set  up  in  the  pole  pieces  during  the  starting 
period,  a  break-up  switch  is  provided  on  the  converter  which  splits 
up  the  shunt  winding  into  a  number  of  sections ;  this  switch  must 
always  be  open  when  the  converter  is  started  from  the  alternating 
current  side  and  is  closed  after  the  machine  has  attained  synchro- 
nous speed. 

The  connections  of  the  auxiliary  appliances,  such  as  instru- 
ments, etc.,  on  the  alternating  side  require  little  explanation,  as 
they  are  apparent  from  the  figure.  When  the  converter  is  started 
from  the  direct-current  side,  it  has  to  be  synchronized  before  the 
main  switch  on  the  alternating-current  side  is  closed ;  it  is  there- 
fore necessary  to  provide  devices  for  indicating  synchronism  and 
in  this  case  synchronizing  lamps  or  the  voltmeter  are  used  as  ex- 
plained in  Art.  185.  The  ammeter  which  indicates  the  alternating 
current  supplied  to  the  converter  is  not  connected  directly  in  the 
main  conductors,  but  is  operated  from  the  secondaries  of  series 
transformers  as  shown.  These  transformers  furnish  a  current 
proportional  to  the  main  current  and  completely  separate  the  in- 
struments, operated  by  them,  from  the  high-tension  lines.  The 
current  from  these  transformers  passes  through  the  coils  of  the 
overload  relays,  and  if  the  main  current  for  any  reason  becomes 
excessive,  either  one  or  both  of  these  relays  opens  the  secondary 
circuit,  so  that  the  current  from  the  transformers  flows  through 
the  tripping  coil  of  the  main  switch,  thus  causing  the  switch  to 
open  and  disconnect  the  converter  from  the  system. 

The  connections  on  the  direct-current  side  are  very  similar  to 
those  for  an  ordinary  direct-current,  compound-wound,  dynamo 
operating  on  a  street-railway  system.  The  negative  side  is  con- 
nected to  the  rail  and  the  positive  side  to  the  feeders  supplying 
the  trolley  wire  or  third  rail.  When  a  number  of  machines  are 
operated  in  parallel  it  is  necessary  that  their  series  field  coils  also 
be  connected  in  parallel  in  order  to  secure  a  proper  equalization 
of  load  between  the  machines.  The  negative  brushes  (i.  e.,  the 


ROTARY  CONVERTERS  AND  MOTOR  GENERATORS.  343 

brushes  to  which  in  this  case  the  series  coils  are  attached)  are, 
therefore,  connected  by  way  of  the  equalizer  switch,  to  the  equal- 
izing bus-bar  which  is  usually  located  near  the  converters  rather 
than  on  the  switchboard.  Very  often  compound-wound  machines 
are  equalized  on  the  +  side  instead  of  the  —  side,  i.  e.,  the  series 
coils  and  equalizer  connections  are  attached  to  the  positive 
brushes.  .  Either  way  is  equally  effective,  but  the  arrangement 
shown  in  Fig.  250  is  well  adapted  for  street  railway  sub-stations, 
because  the  rail  or  —  bus  bar  and  the  equalizer  bar  are  at  prac- 
tically the  same  potential  and  can  be  located  together  in  close 
proximity  to  the  converters.  Only  the  +  cables  need  be  run 
to  the  switchboard  and  the  connections  are  thus  simplified  and 
shortened.  The  direct-current  side  of  each  converter  is  equipped 
with  a  starting  rheostat  switch  connected  across  the  main  positive 
switch  as  shown.  When  the  converter  is  started  by  means  of 
direct  current,  obtained  from  the  direct  current  bus  bars,  the 
main  -(-  switch  is  open  and  all  the  starting  current  has  to  flow 
through  the  starting  resistance.  This  limits  the  rush  of  current 
and  as  the  converter  comes  up  to  speed,  the  rheostat  switch  is 
pushed  in,  thus  cutting  out  the  resistance.  After  the  converter 
has  been  started  it  can  be  synchronized  by  adjusting  the  field 
strength,  and  hence  the  speed,  until  the  voltmeter  or  other 
synchronism  indicator  shows  that  the  converter  is  in  step. 


THE   END. 


INDEX. 


NUMBERS    RELATE   TO    PAGES. 


ADMITTANCE,  definition  of,  81 

All-day  efficiency,  151. 

Alternating  current,  definition  of,   19. 

current  and  electromotive  force, 
representation  of,  by  polar  co- 
ordinates, 27. 

current  and  electromotive  force, 
representation  of,  by  rectangu- 
lar coordinates,  25. 

current  problems,  characteristic 
features  of,  24. 

currents,  advantages  and  disad- 
vantages of,  22. 

electromotive  force,  definition  of, 

19. 
Alternator    armatures,    types    of,    126. 

Bullock,  280,  281. 

characteristic  curve  of,  120. 

the  constant  current,   121. 

design,   135. 

fundamental   equation   of,    30. 

the  hydraulic  analogue  of,  21. 

the  inductor,   20. 

output,  limits  of,  124. 

the  simple,   19. 

speeds,   126. 

the  three-phase,    102. 

the  two-phase,   98. 
Alternators,   273. 

field  excitation  of,   137. 

inductor,   274,  281. 

parallel    running    of,    196,    283. 

polyphase,  98. 

rotating  armature,   274,    275; 

rotating  field,  274,  277. 
Ammeter,     the     electro-dynamometer, 
37- 


Ammeter,  hot  wire,  37. 

plunger  type,  41. 
Ammeters,    37. 
Analogies,   mechanical   and   electrical, 

15- 
Armature    drop,    118. 

inductance,    116. 

insulation,  133. 

reaction   in   alternators,    114. 

windings,  127. 

windings,  single-phase,    128. 

windings,  three-phase,    133. 

windings,  two-phase,   129. 
Autotransformer,    the,    158. 

high  duty  of,  160. 

Average   values    and   effective   values, 
29. 

values  of  harmonic  electromotive 
force  and  current,  57. 

CAPACITY,   calculation  of,    15. 
electrostatic,    13. 
of  transmission  lines,  264. 
units   of,    13. 
Compensators,   starting,   for  induction 

motors,    309. 

Compensated  alternator,  the,  140. 
Complex    quantity,    application   of,   to 

alternating  currents,  80. 
quantity  and  simple  quantity,  76. 
Compound  field  windings,   138. 
Composite    field    windings,    see    com- 
pound  field  windings. 
Composition    and    resolution    of    har- 
monic     electromotive      forces      and 
currents,   52. 
Concentrated  windings,  21. 


345 


346 


INDEX. 


Condensers,    13. 

Condenser,    hydrostatic    analogue    of, 

14. 

Conductance,   definition   of,   81. 
Connections   for  converter  substation, 

339- 

Contact-maker,  the,  32. 
Converter,   210,   213. 

armature  current  of,  216. 

armature  heating  of,  225. 

armature  reaction  of,  217. 

current  relations  of,   221. 

electromotive  relations  of,  219. 

hunting  of,  217. 

instantaneous  armature  current, 
222. 

rating  of,  218. 

six-phase,  connections  for,  333. 

starting  of,  215,   328. 

use  of,    214. 

use  of,  to  supply  current  to  Edi- 
son three-wire  system,  226. 
Converters,    326. 

behavior  of  when  converting 
from  direct  to  alternating  cur- 
rent, 330. 

construction   of,   326. 

frequencies  of,   334. 

starting  of,   328. 

transformer  connections  for,  331. 

voltage  regulation  of,  334. 
Copper  losses   of  transformer,   150. 
Core  flux  of  transformer,  148. 

losses   of  transformer,    150. 
Coulomb,  definition  of,   12. 
Current   curves,   25. 

decay  of,  9,  10. 

densities  in  armatures,   135. 

growth  of,  9,  ii. 
Cycle,   definition   of,   24,   51. 

DECAY  of  current,  9,  10. 
Distributed  windings,   21. 

EFFECTIVE  values  of  harmonic  electro- 
motive force  and  current,   58. 
Electric   charge,    definition   of,    12. 


Electric  charge,  units  of,  12. 

current,  kinetic  energy  of,  3. 
Electrical    and    mechanical    analogies, 

15- 

Electrodynamometer,  the,  37. 
Electrometer,    absolute,    40. 
Electromotive      force      and      current 

curves,   determination   of,   32. 
force    loss   in   transmission    lines, 

263. 
force  required  to  make  a  current 

change,  6. 
force  curves,  25. 

Electrostatic  capacity,  see  capacity. 
Effective   values   and   average    values, 

29. 

Electric   charge,   measurement   of,    12. 
Energy    flow    in    polyphase    systems, 

1 10. 

kinetic,  of  the  electric  current,  3. 

Equivalent  inductance  of  magnetic 
leakage  of  transformer,  175. 

Excitation  characteristics  of  synchro- 
nous motor,  207. 

Exciter,  the  field,  21. 

FARAD,  definition  of,  13. 

Field,   distortion   in   alternators,    114. 

Flux,  magnetic,   i. 

Flux-turns,  definition  of,  6. 

Form-factor,  definition  of,  29. 

factor  of  harmonic  electromotive 

force,   59. 

Frequencies,   customary,   125. 
Frequency,  definition  of,  24,  51. 

relation  of  to  speed,  24. 

GROWTH   of  current,  9,    n. 

HARMONIC  current  in  inductive  cir- 
cuit, Problem  IV.,  63. 

current  in  inductive  circuit  con- 
taining a  condenser,  Problem 
VI.,  66. 

current,  rate  of  change  of,  55. 

electromotive  force  and  current, 
average  values  of,  57. 


INDEX. 


347 


Harmonic     electromotive     force     and 

current,  definition  of,  49. 
electromotive    force   and   current, 

effective  values  of,  58. 
electromotive    force,    form    factor 

of,    59- 
electromotive     force    or    current, 

algebraic  expression  for,  50. 
electromotive    force    or    current, 
representation    of,    by    rotating 
vector,  49. 

Harmonic  electromotive  forces  and 
currents,  composition  and  resolu- 
tion of,  52. 

Henry,  definition  of  the,  4. 
Heyland    diagram     of    the    induction 

motor,  257. 

Hunting  of  synchronous  motor,   198. 
Hunting,  prevention  of,  325. 

IMPEDANCE,  definition  of,  80. 
Inductance,  calculation  of,  8. 
definition  of,  3. 
dependence  of,  upon  size,   5. 
dependence  of,  upon  turns,  5. 
mechanical  analogue  of,  5. 
units  of,  4. 

[nduction  generator,   240. 
motor,  230. 

motor,  complex  equations  of,  249. 
motor,  direction    of    rotation    of, 

316. 
motor,  effect  of  eddy  currents  and 

hysteresis,  247. 
motor,  effect  of  magnetic  leakage, 

248. 
motor,  effect  of  stator  resistance, 

247. 

motor,  general  theory  of,  242. 
motor,  graphical  solution  of,  256. 
motor,  maximum   torque   of,   255. 
motor,  simple  discussion  of,  237. 
motor,  single-phase    driving,    240. 
motor,  single-phase,  316. 
motor,  speed  regulation  of,  311. 
motor,  starting  compensator,  309. 


Inductance  motor,   starting  torque  of, 
255. 

motor,   torque-speed  relations,  238. 

motor,  Vector   diagram   of,   245. 

motor,  Wagner  single-phase,  319. 

motors,  305. 

motors,  current    consumption    of, 
314. 

motors,  frequencies  for,  315. 

motors,    standard     voltages     for, 
314- 

motors,     transformers  for,  315. 

wattmeter,  the,  241. 
Inductivities,  table  of,    14. 
Inductivity,  definition  of,  14. 
Inductor,   alternator,  the,   20,   274. 

LAGGING    currents,    compensation    for, 

91. 
Lincoln  synchronizer.  287. 

MAGNETIC  flux,  i. 

flux    densities    in    armature    and 

in  air  gap,  134. 
Mechanical  analogue  of  inductance,  5. 

and  electrical  analogies,  15. 
Microfarad,   definition  of,    13. 
Monocyclic  system,  the,  164. 
Motor-generators,  336. 
Motor,  induction,  see  induction  motor. 

repulsion,  317. 

the  synchronous,  see  synchronous 
motor. 

NON-INDUCTIVE  circuits,  4. 

OPPOSITION,  definition  of,  52. 
Output  of  alternator,  limits  of,   124. 
of  transformer,  limits  of,  152. 

PHASE  constant  of  armature  winding, 
122. 

difference,   definition  of,  51. 
Polyphase    systems,    98. 

systems,  transformers  for,   161. 

transformers,  302. 


348 


INDEX. 


Power,  average,  27. 

expression  for,  81. 

expression  for  in  case  of  har- 
monic electromotive  force  and 
current,  59. 

factor,  definition  of,   60. 

in  polyphase  systems,   109. 

instantaneous,  27. 

loss  in  transmission  lines,  263. 

measurements  in  polyphase  sys- 
tems, in. 

measurement,  three  ammeter 
method,  44. 

measurement,       three       voltmeter 

method,   43. 

Prevention  of  hunting,  325. 
Problem  L,  10. 

II.,  ii. 

III.,  harmonic  current  in  non-in- 
ductive circuit,  63. 

IV.,  harmonic  current  in  induc- 
tive circuit,  63. 

V.,  establishment  of  harmonic 
current  in  inductive  current,  65. 

VI.,  harmonic  current  in  induc- 
tive circuit  containing  a  con- 
denser, 66. 

VII.,  coils  in  series,  84. 

VIIL,   coils  in  parallel,   88. 
Pumping    of    synchronous    motor,    see 
hunting  of  synchronous  motor. 

QUADRATURE,  definition  of,  51. 
Quantity,  simple  and  complex,   76. 

REACTANCE,  definition  of,  80. 

negative  of  synchronous  motor, 
209. 

of  transmission  lines,  263. 
Reaction  of  changing  current,  7. 
Recording  wattmeter,  47,  241. 
Rectifier,  the  alternating  current,  30. 
Repulsion  motor,  317. 
Resonance,   electric,   definition   of,   69. 

electric,  mechanical,  analogues  of, 
71,  72. 


Resonance,    multiplication    of    current 

by,  71. 
multiplication      of      electromotive 

force  by,    71. 

Rice's   compensated  alternator,    140. 
Rotary  converter,  see  converter. 
Rotary  converter,  sub-station  connec- 
tions for,  339. 
Rotor,   squirrel-cage,   230. 

SCOTT'S  transformer,   163. 
Self-induced  electromotive  force,  7. 
Self-induction,    coefficient   of,   see   in- 
ductance. 

Simple  quantity  and  complex  quan- 
tity, 76. 

Single-phase  induction  motor,  316. 
Spark  gauge,  the,  41. 
Speeds  of  alternators,  126. 
Squirrel-cage  rotor,  230. 
Stator  of  induction  motor,   230. 

windings,  231. 

Steinmetz's  method  of  representing 
harmonic  electromotive  force  and 
current,  60. 

Step-up  and  step-down  transforma- 
tion, 144. 

Sub-stations,  connections  for,  339. 
Susceptance,  definition  of,  81. 
Synchronism,  definition  of,  51. 
Synchronizer,    Lincoln,    287. 
Synchronous  motor,  186,  311. 
motor,  advantages  of,  324. 
motor,  conditions     of     maximum 

efficiency,   200. 

motor,  construction    of,    322. 
Synchronous    motor,    excitation    char- 
acteristics  of,   207. 
motor,    greatest     possible     intake 

of,  206. 

motor,  hunting   of,    198. 
motor,  maximum  intake  of,  205. 
motor,  necessity   of   synchronism, 

191. 

motor,  negative  reactance  of,  209. 
motor,  the  polyphase,  210. 
motor,  stability   of,    195. 


INDEX. 


349 


Synchronous  motor,  starting  of,    194, 

322. 

motor,  stoppage  due  to  overload, 
191. 

THOMSON   inclined  coil  meters,  42. 

recording  wattmeter,  47. 
Three-ammeter  method  for  power,  44. 
Three-phase  alternator,   the,    102. 
Thiee-phase  systems,   106. 

transformer   connections,   299. 
Three-voltmeter  method  for  power,  43. 
Transformation,   ratio   of,    145. 

step-up    and    step-down,    144. 
Transformer,   the,   143. 

action,    144. 

action,    effect    of    coil    resistance, 

173- 

action,  effect  of  magnetic  leak- 
age, 174. 

admittance  of,  at  zero  output, 
170. 

all-day  efficiency  of,  151. 

complex  equations  of,   180. 

connections,   156. 

connections  for  converters,  331. 

connections,  three-phase,  299. 

connections,  two-phase,    297. 

the  constant  current,  176,  303. 

core  flux  of,  148. 

design,   153. 

efficiencies,  table  of,   151. 

efficiency   of,    150. 

equivalent  resistance  and  react- 
ance of,  147. 

graphical  solution  of,  256. 

leakage   inductance   of,    175,    177. 

losses,    149. 

magnetic   leakage   of,    169. 

magnetizing  current  of,   170,  171. 

output,   limits    of,    152. 

rating,    152. 

regulation,   172. 

regulation,   calculation   of,    183. 

the   Scott,    163. 

two-phase  three-phase,    162. 

the,  without  iron,  92. 


Transformers,    291. 
cooling  of,  294. 
for  induction  motors,   315. 
for  polyphase  systems,  261,  302. 
with  divided  coils,  157. 
Transmission  line  calculation,  266. 
lines,  262. 
lines,  interference      of     separate, 

265. 

Three-phase  alternator,  102. 
Three-phase  electromotive   forces  and 

currents,    103. 

Three-phase,   three-wire  system,   elec- 
tromotive    force     and     current 
relations    in,    106,    107. 
transformer      connections,       161, 

299. 

Two-phase  alternator,    98. 
Two-phase  electromotive    forces    and 

currents,  101. 

three-phase   transformer,    162. 
three-wire   system,    effect   of  line 

drop   in,    101. 

three-wire  system  electromotive 
force  and  current  relations  in, 
I  OX. 

transformer  connections,  161, 
297. 

VECTORS,  77. 

addition    and    subtraction    of,    78. 

division  of,   79. 

multiplication   of,   78. 
Voltage  regulators,   334. 
Voltmeters,    37. 

electrodynamometer,   39. 

electrostatic,  40. 

hot   wire,    37. 

plunger,   type,  41. 

WAGNER  single-phase  induction  motor, 

319- 
Wattmeter,    37,   44. 

inductance    error    of,    94- 
the  induction,    241. 
the  recording,  47. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


1719 


LD  21-100m-7,'52(A2528sl6)476 


Yc  32602 


..«. 
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